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Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver, Chehrzad Shakiban

Chapter 3

Inner Products and Norms - all with Video Answers

Educators

Chapter Questions

03:15

Problem 1

Prove that the formula $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1-v_1 w_2-v_2 w_1+b v_2 w_2$ defines an inner product on $\mathbb{R}^2$ if and only if $b>1$.

Wendi Zhao
Wendi Zhao
Numerade Educator
04:58

Problem 1

Verify the Cauchy-Schwarz inequality for each of the following pairs of vectors $\mathbf{v}, \mathbf{w}$, using the standard dot product, and then determine the angle between them:
(a) $(1,2)^T,(-1,2)^T$,
(b) $(1,-1,0)^T,(-1,0,1)^T$,
(c) $(1,-1,0)^T,(2,2,2)^T$,
(d) $(1,-1,1,0)^T,(-2,0,-1,1)^T$,
(e) $(2,1,-2,-1)^T,(0,-1,2,-1)^T$.

Ahmad Reda
Ahmad Reda
Numerade Educator
01:46

Problem 1

Compute the $1,2,3$, and $\infty$ norms of the vectors $\left(\begin{array}{l}1 \\ 0\end{array}\right),\left(\begin{array}{l}0 \\ 1\end{array}\right)$. Verify the triangle inequality in each case.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:30

Problem 1

Which of the following $2 \times 2$ matrices are positive definite?
(a) $\left(\begin{array}{ll}1 & 0 \\ 0 & 2\end{array}\right)$,
(b) $\left(\begin{array}{ll}0 & 1 \\ 2 & 0\end{array}\right)$,
(c) $\left(\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rr}5 & 3 \\ 3 & -2\end{array}\right)$,
(e) $\left(\begin{array}{rr}1 & -1 \\ -1 & 3\end{array}\right)$
(f) $\left(\begin{array}{rr}1 & 1 \\ -1 & 2\end{array}\right)$
In the positive definite cases, write down the formula for the associated inner product.

Chris Trentman
Chris Trentman
Numerade Educator
02:30

Problem 1

Are the following matrices are positive definite?
(a) $\left(\begin{array}{rr}4 & -2 \\ -2 & 4\end{array}\right)$,
(b) $\left(\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right)$
(c) $\left(\begin{array}{lll}1 & 1 & 2 \\ 1 & 2 & 1 \\ 2 & 1 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}1 & 1 & 1 \\ 1 & 2 & -2 \\ 1 & -2 & 4\end{array}\right)$,
(e) $\left(\begin{array}{llll}2 & 1 & 1 & 1 \\ 1 & 2 & 1 & 1 \\ 1 & 1 & 2 & 1 \\ 1 & 1 & 1 & 2\end{array}\right)$,
(f)
$\left(\begin{array}{rrrr}-1 & 1 & 1 & 1 \\ 1 & -1 & 1 & 1 \\ 1 & 1 & -1 & 1 \\ 1 & 1 & 1 & -1\end{array}\right)$

Chris Trentman
Chris Trentman
Numerade Educator
02:08

Problem 1

Write down a single equation that relates the five most important numbers in mathematics, which are $0,1, e, \pi$, and $\mathrm{i}$.

Chris Trentman
Chris Trentman
Numerade Educator
03:14

Problem 2

Which of the following formulas for $\langle\mathbf{v}, \mathbf{w}\rangle$ define inner products on $\mathbb{R}^2$ ?
(a) $2 v_1 w_1+3 v_2 w_2$, (b) $v_1 w_2+v_2 w_1$,
(c) $\left(v_1+v_2\right)\left(w_1+w_2\right)$,
(d) $v_1^2 w_1^2+v_2^2 w_2^2$,
(e) $\sqrt{v_1^2+v_2^2} \sqrt{w_1^2+w_2^2}$,
(f) $2 v_1 w_1+\left(v_1-v_2\right)\left(w_1-w_2\right)$,
(g) $4 v_1 w_1-2 v_1 w_2-2 v_2 w_1+4 v_2 w_2$.

Wendi Zhao
Wendi Zhao
Numerade Educator
02:52

Problem 2

(a) Find the Euclidean angle between the vectors $(1,1,1,1)^T$ and $(1,1,1,-1)^T$ in $\mathbb{R}^4$. (b) List the possible angles between $(1,1,1,1)^T$ and $\left(a_1, a_2, a_3, a_4\right)^T$, where each $a_i$ is either 1 or -1 .

Lucas Finney
Lucas Finney
Numerade Educator
01:11

Problem 2

Answer Exercise 3.3 .1 for (a) $\left(\begin{array}{r}2 \\ -1\end{array}\right),\left(\begin{array}{r}1 \\ -2\end{array}\right)$,
(b) $\left(\begin{array}{r}1 \\ 0 \\ -1\end{array}\right),\left(\begin{array}{r}-1 \\ 1 \\ 0\end{array}\right)$,
, (c) $\left(\begin{array}{r}1 \\ -2 \\ -1\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ -3\end{array}\right)$.

Wendy Wang
Wendy Wang
Numerade Educator
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Problem 2

Let $K=\left(\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right)$. Prove that the associated quadratic form $q(\mathbf{x})=\mathbf{x}^T K \mathbf{x}$ is indefinite by finding a point $\mathbf{x}^{+}$where $q\left(\mathbf{x}^{+}\right)>0$ and a point $\mathbf{x}^{-}$where $q\left(\mathbf{x}^{-}\right)<0$.

Victor Salazar
Victor Salazar
Numerade Educator
03:10

Problem 2

Find an $L D L^T$ factorization of the following symmetric matrices. Which are positive
definite?
(a) $\left(\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right)$,
(b) $\left(\begin{array}{rr}5 & -1 \\ -1 & 3\end{array}\right)$,
(c) $\left(\begin{array}{rrr}3 & -1 & 3 \\ -1 & 5 & 1 \\ 3 & 1 & 5\end{array}\right)$,
(d) $\left(\begin{array}{rrr}-2 & 1 & -1 \\ 1 & -2 & 1 \\ -1 & 1 & -2\end{array}\right)$,
(e) $\left(\begin{array}{rrr}2 & 1 & -2 \\ 1 & 1 & -3 \\ -2 & -3 & 11\end{array}\right)$,
(f) $\left(\begin{array}{llll}1 & 1 & 1 & 0 \\ 1 & 2 & 0 & 1 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 2\end{array}\right)$,
(g) $\left(\begin{array}{llll}3 & 2 & 1 & 0 \\ 2 & 3 & 0 & 1 \\ 1 & 0 & 3 & 2 \\ 0 & 1 & 2 & 3\end{array}\right)$,
(h) $\left(\begin{array}{rrrr}2 & 1 & -2 & 0 \\ 1 & 1 & -3 & 2 \\ -2 & -3 & 10 & -1 \\ 0 & 2 & -1 & 7\end{array}\right)$.

Victor Salazar
Victor Salazar
Numerade Educator
01:33

Problem 2

For any integer $k$, prove that $e^{k \pi \mathrm{i}}=(-1)^k$.

Clarissa Noh
Clarissa Noh
Numerade Educator
02:20

Problem 3

Show that $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1+v_1 w_2+v_2 w_1+v_2 w_2$ does not define an inner product on $\mathbb{R}^2$.

Wendi Zhao
Wendi Zhao
Numerade Educator
00:49

Problem 3

Prove that the points $(0,0,0),(1,1,0),(1,0,1),(0,1,1)$ form the vertices of a regular tetrahedron, meaning that all sides have the same length. What is the common Euclidean angle between the edges? What is the angle between any two rays going from the center $\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)$ to the vertices? Remark. Methane molecules assume this geometric configuration, and the angle influences their chemistry.

Victor Salazar
Victor Salazar
Numerade Educator
01:02

Problem 3

Which two of the vectors $\mathbf{u}=(-2,2,1)^T, \mathbf{v}=(1,4,1)^T, \mathbf{w}=(0,0,-1)^T$ are closest to each other in distance for (a) the Euclidean norm? (b) the $\infty$ norm? (c) the 1 norm?

Manik Pulyani
Manik Pulyani
Numerade Educator
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Problem 3

(a) Prove that a diagonal matrix $D=\operatorname{diag}\left(c_1, c_2, \ldots, c_n\right)$ is positive definite if and only if all its diagonal entries are positive: $c_i>0 . \quad(b)$ Write down and identify the associated inner product.

Victor Salazar
Victor Salazar
Numerade Educator
01:51

Problem 3

(a) For which values of $c$ is the matrix $A=\left(\begin{array}{lll}1 & 1 & 0 \\ 1 & c & 1 \\ 0 & 1 & 1\end{array}\right)$ positive definite? (b) For the particular value $c=3$, carry out elimination to find the factorization $A=L D L^T$. (c) Use your result from part $(b)$ to rewrite the quadratic form $q(x, y, z)=x^2+2 x y+3 y^2+2 y z+z^2$ as a sum of squares. (d) Explain how your result is related to the positive definiteness of $A$.

Srilakshmi E K
Srilakshmi E K
Numerade Educator
05:52

Problem 3

Is the formula $1^z=1$ valid for all complex values of $z$ ?

Uma Kumari
Uma Kumari
Numerade Educator
03:14

Problem 4

Prove that each of the following formulas for $\langle\mathbf{v}, \mathbf{w}\rangle$ defines an inner product on $\mathbb{R}^3$. Verify all the inner product axioms in careful detail:
(a) $v_1 w_1+2 v_2 w_2+3 v_3 w_3$,
(b) $4 v_1 w_1+2 v_1 w_2+2 v_2 w_1+4 v_2 w_2+v_3 w_3$,
(c) $2 v_1 w_1-2 v_1 w_2-2 v_2 w_1+3 v_2 w_2-v_2 w_3-v_3 w_2+2 v_3 w_3$.

Wendi Zhao
Wendi Zhao
Numerade Educator
03:08

Problem 4

Verify the Cauchy-Schwarz inequality for the vectors $\mathbf{v}=(1,2)^T, \mathbf{w}=(1,-3)^T$, using (a) the dot product; (b) the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1+2 v_2 w_2$; (c) the inner product (3.9).

Anthony Ramos
Anthony Ramos
Numerade Educator

Problem 4

(a) Compute the $\mathrm{L}^{\infty}$ norm on $[0,1]$ of the functions $f(x)=\frac{1}{3}-x$ and $g(x)=x-x^2$.
(b) Verify the triangle inequality for these two particular functions.

Check back soon!
04:34

Problem 4

Write out the Cauchy-Schwarz and triangle inequalities for the inner product defined in Example 3.28.

Ahmad Reda
Ahmad Reda
Numerade Educator
View

Problem 4

Write the quadratic form $q(\mathbf{x})=x_1^2+x_1 x_2+2 x_2^2-x_1 x_3+3 x_3^2$ in the form $q(\mathbf{x})=\mathbf{x}^T K \mathbf{x}$ for some symmetric matrix $K$. Is $q(\mathbf{x})$ positive definite?

Victor Salazar
Victor Salazar
Numerade Educator
01:20

Problem 4

What is wrong with the calculation $e^{2 a \pi \mathrm{i}}=\left(e^{2 \pi \mathrm{i}}\right)^a=1^a=1$ ?

Linh Vu
Linh Vu
Numerade Educator
00:56

Problem 5

The unit circle for an inner product on $\mathbb{R}^2$ is defined as the set of all vectors of unit length: $\|\mathbf{v}\|=1$. Graph the unit circles for (a) the Euclidean inner product, $(b)$ the weighted inner product (3.8), (c) the non-standard inner product (3.9). (d) Prove that cases $(b)$ and $(c)$ are, in fact, both ellipses.

Wendi Zhao
Wendi Zhao
Numerade Educator
03:08

Problem 5

Verify the Cauchy-Schwarz inequality for the vectors $\mathbf{v}=(3,-1,2)^T, \mathbf{w}=(1,-1,1)^T$, using (a) the dot product; (b) the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1+2 v_2 w_2+3 v_3 w_3$; (c) the inner product $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T\left(\begin{array}{rrr}2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right) \mathbf{w}$.

Anthony Ramos
Anthony Ramos
Numerade Educator
00:49

Problem 5

Answer Exercise 3.3.4 using the $\mathrm{L}^1$ norm.

Karly Williams
Karly Williams
Numerade Educator
03:08

Problem 5

(a) Show that every diagonal entry of a positive definite matrix must be positive.
(b) Write down a symmetric matrix with all positive diagonal entries that is not positive definite. (c) Find a nonzero matrix with one or more zero diagonal entries that is positive semi-definite.

ET
Ed Tam
Numerade Educator
02:22

Problem 5

Write the following quadratic forms on $\mathbb{R}^2$ as a sum of squares. Which are positive definite?
(a) $x^2+8 x y+y^2$,
(b) $x^2-4 x y+7 y^2$,
(c) $x^2-2 x y-y^2$,
(d) $x^2+6 x y$.

Victor Salazar
Victor Salazar
Numerade Educator
02:00

Problem 5

(a) Write $\mathrm{i}$ in phase-modulus form. (b) Use this expression to find $\sqrt{\mathrm{i}}$, i.e., a complex number $z$ such that $z^2=\mathrm{i}$. Can you find a second square root? (c) Find explicit formulas for the three third roots and four fourth roots of $i$.

Ruchika Sangwan
Ruchika Sangwan
Numerade Educator
00:51

Problem 6

(a) Explain why the formula for the Euclidean norm in $\mathbb{R}^2$ follows from the Pythagorean Theorem. (b) How do you use the Pythagorean Theorem to justify the formula for the Euclidean norm in $\mathbb{R}^3$ ?

Amy Jiang
Amy Jiang
Numerade Educator
01:28

Problem 6

Show that one can determine the angle $\theta$ between $\mathbf{v}$ and $\mathbf{w}$ via the formula $\cos \theta=\frac{\|\mathbf{v}+\mathbf{w}\|^2-\|\mathbf{v}-\mathbf{w}\|^2}{4\|\mathbf{v}\|\|\mathbf{w}\|}$. Draw a picture illustrating what is being measured.

Vipender Yadav
Vipender Yadav
Numerade Educator
01:17

Problem 6

Which two of the functions $f(x)=1, g(x)=x, h(x)=\sin \pi x$ are closest to each other on the interval $[0,1]$ under (a) the $\mathrm{L}^1$ norm?
(b) the $\mathrm{L}^2$ norm?
(c) the $\mathrm{L}^{\infty}$ norm?

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:00

Problem 6

Prove that if $K$ is any positive definite matrix, then every positive scalar multiple $c K$, $c>0$, is also positive definite.

Chris Trentman
Chris Trentman
Numerade Educator
05:23

Problem 6

Prove that the following quadratic forms on $\mathbb{R}^3$ are positive definite by writing each as a sum of squares: (a) $x^2+4 x z+3 y^2+5 z^2$, (c) $2 x_1^2+x_1 x_2-2 x_1 x_3+2 x_2^2-2 x_2 x_3+2 x_3^2$.
(b) $x^2+3 x y+3 y^2-2 x z+8 z^2$,

Victor Salazar
Victor Salazar
Numerade Educator
02:36

Problem 6

In Figure 3.7 , where would you place the point $1 / z$ ?

K Joseph
K Joseph
Numerade Educator
02:36

Problem 7

Prove that the norm on an inner product space satisfies $\|c \mathbf{v}\|=|c|\|\mathbf{v}\|$ for every scalar $c$ and vector $\mathbf{v}$.

Carson Merrill
Carson Merrill
Numerade Educator
09:29

Problem 7

The Law of Cosines: Prove that the formula
$$
\|\mathbf{v}-\mathbf{w}\|^2=\|\mathbf{v}\|^2+\|\mathbf{w}\|^2-2\|\mathbf{v}\|\|\mathbf{w}\| \cos \theta,
$$
where $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{w}$, is valid in every inner product space.

KS
Kristan Siegel
Numerade Educator
08:24

Problem 7

Consider the functions $f(x)=1$ and $g(x)=x-\frac{3}{4}$ as elements of the vector space $\mathrm{C}^0[0,1]$. For each of the following norms, compute $\|f\|,\|g\|,\|f+g\|$, and verify the triangle inequality: (a) the $\mathrm{L}^1$ norm; (b) the $\mathrm{L}^2$ norm; (c) the $\mathrm{L}^3$ norm; (d) the $\mathrm{L}^{\infty}$ norm.

Matthew Allcock
Matthew Allcock
Numerade Educator
03:00

Problem 7

(a) Show that if $K$ and $L$ are positive definite matrices, so is $K+L$. (b) Give an example of two matrices that are not positive definite whose sum is positive definite.

Chris Trentman
Chris Trentman
Numerade Educator
02:27

Problem 7

Write the following quadratic forms in matrix notation and determine if they are positive definite:
(a) $x^2+4 x z+2 y^2+8 y z+12 z^2$,
(b) $3 x^2-2 y^2-8 x y+x z+z^2$,
(c) $x^2+2 x y+2 y^2-4 x z-6 y z+6 z^2$,
(d) $3 x_1^2-x_2^2+5 x_3^2+4 x_1 x_2-7 x_1 x_3+9 x_2 x_3$,
(e) $x_1^2+4 x_1 x_2-2 x_1 x_3+5 x_2^2-2 x_2 x_4+6 x_3^2-x_3 x_4+4 x_4^2$.

Victor Salazar
Victor Salazar
Numerade Educator
03:36

Problem 7

(a) If $z$ moves counterclockwise around a circle of radius $r$ in the complex plane, around which circle and in which direction does $w=1 / z$ move?
(b) What about $w=\bar{z}$ ?
(c) What if the circle is not centered at the origin?

Uma Kumari
Uma Kumari
Numerade Educator
04:18

Problem 8

Prove that $\langle a \mathbf{v}+b \mathbf{w}, c \mathbf{v}+d \mathbf{w}\rangle=a c\|\mathbf{v}\|^2+(a d+b c)\langle\mathbf{v}, \mathbf{w}\rangle+b d\|\mathbf{w}\|^2$.

Zain Haider
Zain Haider
Numerade Educator
03:11

Problem 8

Use the Cauchy-Schwarz inequality to prove $(a \cos \theta+b \sin \theta)^2 \leq a^2+b^2$ for any $\theta, a, b$.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:12

Problem 8

Answer Exercise 3.3.7 when $f(x)=e^x$ and $g(x)=e^{-x}$.

Christopher Stanley
Christopher Stanley
Numerade Educator
03:00

Problem 8

Find two positive definite matrices $K$ and $L$ whose product $K L$ is not positive definite.

Chris Trentman
Chris Trentman
Numerade Educator
05:23

Problem 8

For what values of $a, b$, and $c$ is the quadratic form $x^2+a x y+y^2+b x z+c y z+z^2$ positive definite?

Victor Salazar
Victor Salazar
Numerade Educator
01:28

Problem 8

Show that $-|z| \leq \operatorname{Re} z \leq|z|$ and $-|z| \leq \operatorname{Im} z \leq|z|$.

Suzanne W.
Suzanne W.
Numerade Educator
02:02

Problem 9

Prove that the second bilinearity formula (3.4) is a consequence of the first and the other two inner product axioms.

Angelo Rendina
Angelo Rendina
Numerade Educator
01:04

Problem 9

Prove that $\left(a_1+a_2+\cdots+a_n\right)^2 \leq n\left(a_1^2+a_2^2+\cdots+a_n^2\right)$ for any real numbers $a_1, \ldots, a_n$. When does equality hold?

Carson Merrill
Carson Merrill
Numerade Educator
01:02

Problem 9

Carefully prove that $\left\|(x, y)^T\right\|=|x|+2|x-y|$ defines a norm on $\mathbb{R}^2$.

Raj Bala
Raj Bala
Numerade Educator

Problem 9

Write down a nonsingular symmetric matrix that is not positive or negative definite.

Check back soon!
01:36

Problem 9

True or false: Every planar quadratic form $q(x, y)=a x^2+2 b x y+c y^2$ can be written as a sum of squares.

Khushbu Rani
Khushbu Rani
Numerade Educator
04:55

Problem 9

Prove that if $\varphi$ is real, then $\operatorname{Re}\left(e^{\mathrm{i} \varphi} z\right) \leq|z|$, with equality if and only if $\varphi=-\operatorname{ph} z$.

John Gehad
John Gehad
Numerade Educator
02:49

Problem 10

Let $V$ be an inner product space. (a) Prove that $\langle\mathbf{x}, \mathbf{v}\rangle=0$ for all $\mathbf{v} \in V$ if and only if $\mathbf{x}=\mathbf{0}$. (b) Prove that $\langle\mathbf{x}, \mathbf{v}\rangle=\langle\mathbf{y}, \mathbf{v}\rangle$ for all $\mathbf{v} \in V$ if and only if $\mathbf{x}=\mathbf{y}$. (c) Let $\mathbf{v}_1, \ldots, \mathbf{v}_n$ be a basis for $V$. Prove that $\left\langle\mathbf{x}, \mathbf{v}_i\right\rangle=\left\langle\mathbf{y}, \mathbf{v}_i\right\rangle, i=1, \ldots, n$, if and only if $\mathbf{x}=\mathbf{y}$.

Jimmy Yao
Jimmy Yao
Numerade Educator
07:28

Problem 10

The cross product of two vectors in $\mathbb{R}^2$ is defined as the scalar
$$
\mathbf{v} \times \mathbf{w}=v_1 w_2-v_2 w_1 \quad \text { for } \quad \mathbf{v}=\left(v_1, v_2\right)^T, \quad \mathbf{w}=\left(w_1, w_2\right)^T .
$$
(a) Does the cross product define an inner product on $\mathbb{R}^2$ ? Carefully explain which axioms are valid and which are not. (b) Prove that $\mathbf{v} \times \mathbf{w}=\|\mathbf{v}\|\|\mathbf{w}\| \sin \theta$, where $\theta$ denotes the angle from $\mathbf{v}$ to $\mathbf{w}$ as in Figure 3.2. (c) Prove that $\mathbf{v} \times \mathbf{w}=0$ if and only if $\mathbf{v}$ and $\mathbf{w}$ are parallel vectors. (d) Show that $|\mathbf{v} \times \mathbf{w}|$ equals the area of the parallelogram defined by $\mathbf{v}$ and $\mathbf{w}$.

Wasim Sher
Wasim Sher
Numerade Educator
06:05

Problem 10

Prove that the following formulas define norms on $\mathbb{R}^2$ : (a) $\|\mathbf{v}\|=\sqrt{2 v_1^2+3 v_2^2}$,
(b) $\|\mathbf{v}\|=\sqrt{2 v_1^2-v_1 v_2+2 v_2^2}$, (c) $\|\mathbf{v}\|=2\left|v_1\right|+\left|v_2\right|$,
(d) $\|\mathbf{v}\|=\max \left\{2\left|v_1\right|,\left|v_2\right|\right\}$,
(e) $\|\mathbf{v}\|=\max \left\{\left|v_1-v_2\right|,\left|v_1+v_2\right|\right\}$,
(f) $\|\mathbf{v}\|=\left|v_1-v_2\right|+\left|v_1+v_2\right|$.

Chris Trentman
Chris Trentman
Numerade Educator
03:20

Problem 10

Let $K$ be a nonsingular symmetric matrix. (a) Show that $\mathbf{x}^T K^{-1} \mathbf{x}=\mathbf{y}^T K \mathbf{y}$, where $K \mathbf{y}=\mathbf{x}$. (b) Prove that if $K$ is positive definite, then so is $K^{-1}$.

Chris Trentman
Chris Trentman
Numerade Educator
04:37

Problem 10

(a) Prove that a positive definite matrix has positive determinant: $\operatorname{det} K>0$.
(b) Show that a positive definite matrix has positive trace: $\operatorname{tr} K>0$. (c) Show that every $2 \times 2$ symmetric matrix with positive determinant and positive trace is positive definite.
(d) Find a symmetric $3 \times 3$ matrix with positive determinant and positive trace that is not positive definite.

Runpeng Li
Runpeng Li
Numerade Educator
02:41

Problem 10

Prove the identities in (3.90) and (3.91).

AG
Ankit Gupta
Numerade Educator
06:35

Problem 11

Prove that $\mathbf{x} \in \mathbb{R}^n$ solves the linear system $A \mathbf{x}=\mathbf{b}$ if and only if
$$
\mathbf{x}^T A^T \mathbf{v}=\mathbf{b}^T \mathbf{v} \quad \text { for all } \quad \mathbf{v} \in \mathbb{R}^m .
$$
The latter is known as the weak formulation of the linear system, and its generalizations are of great importance in the study of differential equations and numerical analysis, [61].

Uma Kumari
Uma Kumari
Numerade Educator
01:04

Problem 11

Explain why the inequality $\langle\mathbf{v}, \mathbf{w}\rangle \leq\|\mathbf{v}\|\|\mathbf{w}\|$, obtained by omitting the absolute value sign on the left-hand side of Cauchy-Schwarz, is valid.

Carson Merrill
Carson Merrill
Numerade Educator
01:34

Problem 11

Which of the following formulas define norms on $\mathbb{R}^3$ ?
(a) $\|\mathbf{v}\|=\sqrt{2 v_1^2+v_2^2+3 v_3^2}$,
(b) $\|\mathbf{v}\|=\sqrt{v_1^2+2 v_1 v_2+v_2^2+v_3^2}$
(c) $\|\mathbf{v}\|=\max \left\{\left|v_1\right|,\left|v_2\right|,\left|v_3\right|\right\}$,
(d) $\|\mathbf{v}\|=\left|v_1-v_2\right|+\left|v_2-v_3\right|+\left|v_3-v_1\right|$,
(e) $\|\mathbf{v}\|=\left|v_1\right|+\max \left\{\left|v_2\right|,\left|v_3\right|\right\}$.

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
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Problem 11

Prove that an $n \times n$ symmetric matrix $K$ is positive definite if and only if, for every $\mathbf{0} \neq \mathbf{v} \in \mathbb{R}^n$, the vectors $\mathbf{v}$ and $K \mathbf{v}$ meet at an acute Euclidean angle: $|\theta|<\frac{1}{2} \pi$.

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 11

(a) Prove that if $K_1, K_2$ are positive definite $n \times n$ matrices, then $K=\left(\begin{array}{cc}K_1 & \mathrm{O} \\ \mathrm{O} & K_2\end{array}\right)$ is a positive definite $2 n \times 2 n$ matrix. (b) Is the converse true?

Victor Salazar
Victor Salazar
Numerade Educator
04:25

Problem 11

Prove $\operatorname{ph}(z / w)=\operatorname{ph} z-\operatorname{ph} w=\operatorname{ph}(z \bar{w})$ is equal to the angle between the vectors representing $z$ and $w$.

Supratim Pal
Supratim Pal
Numerade Educator
12:49

Problem 12

(a) Prove the identity
$$
\langle\mathbf{u}, \mathbf{v}\rangle=\frac{1}{4}\left(\|\mathbf{u}+\mathbf{v}\|^2-\|\mathbf{u}-\mathbf{v}\|^2\right),
$$
which allows one to reconstruct an inner product from its norm. (b) Use (3.11) to find the inner product on $\mathbb{R}^2$ corresponding to the norm $\|\mathbf{v}\|=\sqrt{v_1^2-3 v_1 v_2+5 v_2^2}$.

Chris Trentman
Chris Trentman
Numerade Educator
07:27

Problem 12

Verify the Cauchy-Schwarz inequality for the functions $f(x)=x$ and $g(x)=e^x$ with respect to (a) the $\mathrm{L}^2$ inner product on the interval $[0,1],(b)$ the $\mathrm{L}^2$ inner product on $[-1,1],(c)$ the weighted inner product $\langle f, g\rangle=\int_0^1 f(x) g(x) e^{-x} d x$.

Ahmad Reda
Ahmad Reda
Numerade Educator
03:34

Problem 12

Prove that two parallel vectors $\mathbf{v}$ and $\mathbf{w}$ have the same norm if and only if $\mathbf{v}= \pm \mathbf{w}$.

Vikash Ranjan
Vikash Ranjan
Numerade Educator
01:42

Problem 12

Prove that the inner product associated with a positive definite quadratic form $q(\mathbf{x})$ is given by the polarization formula $\langle\mathbf{x}, \mathbf{y}\rangle=\frac{1}{2}[q(\mathbf{x}+\mathbf{y})-q(\mathbf{x})-q(\mathbf{y})]$.

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 12

Let $\|\cdot\|$ be any norm on $\mathbb{R}^n$. (a) Show that $q(\mathbf{x})$ is a positive definite quadratic form if and only if $q(\mathbf{u})>0$ for all unit vectors, $\|\mathbf{u}\|=1$. (b) Prove that if $S=S^T$ is any symmetric matrix, then $K=S+c$ I $>0$ is positive definite if $c$ is sufficiently large.

Victor Salazar
Victor Salazar
Numerade Educator
00:31

Problem 12

The phase of a complex number $z=x+\mathrm{i} y$ is often written as $\mathrm{ph} z=\tan ^{-1}(y / x)$. Explain why this formula is ambiguous, and does not uniquely define $\mathrm{ph} z$.

AG
Ankit Gupta
Numerade Educator
04:16

Problem 13

(a) Show that, for all vectors $\mathbf{x}$ and $\mathbf{y}$ in an inner product space,
$$
\|\mathbf{x}+\mathbf{y}\|^2+\|\mathbf{x}-\mathbf{y}\|^2=2\left(\|\mathbf{x}\|^2+\|\mathbf{y}\|^2\right) .
$$
(b) Interpret this result pictorially for vectors in $\mathbb{R}^2$ under the Euclidean norm.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:42

Problem 13

Using the $\mathrm{L}^2$ inner product on the interval $[0, \pi]$, find the angle between the functions
(a) 1 and $\cos x$;
(b) 1 and $\sin x$;
(c) $\cos x$ and $\sin x$.

Marcia Chumas
Marcia Chumas
Numerade Educator
00:58

Problem 13

True or false: If $\|\mathbf{v}+\mathbf{w}\|=\|\mathbf{v}\|+\|\mathbf{w}\|$, then $\mathbf{v}, \mathbf{w}$ are parallel vectors.

Monica Miller
Monica Miller
Numerade Educator
01:23

Problem 13

(a) Is it possible for a quadratic form to be positive, $q\left(\mathbf{x}_{+}\right)>0$, at only one point $\mathbf{x}_{+} \in \mathbb{R}^n$ ? (b) Under what conditions is $q\left(\mathbf{x}_0\right)=0$ at only one point?

Nick Johnson
Nick Johnson
Numerade Educator
03:42

Problem 13

Prove that every symmetric matrix $S=K+N$ can be written as the sum of a positive definite matrix $K$ and a negative definite matrix $N$.

Lucas Finney
Lucas Finney
Numerade Educator

Problem 13

Show that if we identify the complex numbers $z, w$ with vectors in the plane, then their Euclidean dot product is equal to $\operatorname{Re}(z \bar{w})$.

Check back soon!
01:21

Problem 14

Suppose $\mathbf{u}, \mathbf{v}$ satisfy $\|\mathbf{u}\|=3,\|\mathbf{u}+\mathbf{v}\|=4$, and $\|\mathbf{u}-\mathbf{v}\|=6$. What must $\|\mathbf{v}\|$ equal? Does your answer depend upon which norm is being used?

Hong Joo Ryoo
Hong Joo Ryoo
Numerade Educator
01:43

Problem 14

Verify the Cauchy-Schwarz inequality for the two particular functions appearing in Exercise 3.1.32 using the $\mathrm{L}^2$ inner product on (a) the unit square; (b) the unit disk.

Ahmad Reda
Ahmad Reda
Numerade Educator
13:18

Problem 14

Prove that the $\infty$ norm on $\mathbb{R}^2$ does not come from an inner product.

Chris Trentman
Chris Trentman
Numerade Educator
02:26

Problem 14

(a) Let $K$ and $L$ be symmetric $n \times n$ matrices. Prove that $\mathbf{x}^T K \mathbf{x}=\mathbf{x}^T L \mathbf{x}$ for all $\mathbf{x} \in \mathbb{R}^n$ if and only if $K=L$. (b) Find an example of two non-symmetric matrices $K \neq L$ such that $\mathbf{x}^T K \mathbf{x}=\mathbf{x}^T L \mathbf{x}$ for all $\mathbf{x} \in \mathbb{R}^n$.

Pagadala Kishore Reddy
Pagadala Kishore Reddy
Numerade Educator
04:54

Problem 14

(a) Prove that every regular symmetric matrix can be decomposed as a linear combination
$$
K=d_1 \mathbf{l}_1 \mathbf{l}_1^T+d_2 \mathbf{l}_2 \mathbf{l}_2^T+\cdots+d_n \mathbf{l}_n \mathbf{1}_n^T
$$
of symmetric rank 1 matrices, as in Exercise 1.8.15, where $\mathbf{l}_1, \ldots, \mathbf{l}_n$ are the columns of the lower unitriangular matrix $L$ and $d_1, \ldots, d_n$ are the pivots, i.e., the diagonal entries of $D$.
(b) Decompose $\left(\begin{array}{rr}4 & -1 \\ -1 & 1\end{array}\right)$ and $\left(\begin{array}{lll}1 & 2 & 1 \\ 2 & 6 & 1 \\ 1 & 1 & 4\end{array}\right)$ in this manner.

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
01:24

Problem 14

(a) Prove that the complex numbers $z$ and $w$ correspond to orthogonal vectors in $\mathbb{R}^2$ if and only if $\operatorname{Re} z \bar{w}=0$.
(b) Prove that $z$ and i $z$ are always orthogonal.

Shafiq Rehman
Shafiq Rehman
Numerade Educator
03:00

Problem 15

Let $A$ be any $n \times n$ matrix. Prove that the dot product identity $\mathbf{v} \cdot(A \mathbf{w})=\left(A^T \mathbf{v}\right) \cdot \mathbf{w}$ is valid for all vectors $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$.

Joe Lesueur
Joe Lesueur
Numerade Educator
View

Problem 15

(a) Find $a$ such that $(2, a,-3)^T$ is orthogonal to $(-1,3,-2)^T$.
(b) Is there any value of $a$ for which $(2, a,-3)^T$ is parallel to $(-1,3,-2)^T$ ?

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 15

Can formula (3.11) be used to define an inner product for (a) the 1 norm $\|\mathbf{v}\|_1$ on $\mathbb{R}^2$ ?
(b) the $\infty$ norm $\|\mathbf{v}\|_{\infty}$ on $\mathbb{R}^2$ ?

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 15

Suppose $q(\mathbf{x})=\mathbf{x}^T A \mathbf{x}=\sum_{i, j=1}^n a_{i j} x_i x_j$ is a general quadratic form on $\mathbb{R}^n$, whose coefficient matrix $A$ is not necessarily symmetric. Prove that $q(\mathbf{x})=\mathbf{x}^T K \mathbf{x}$, where $K=\frac{1}{2}\left(A+A^T\right)$ is a symmetric matrix. Therefore, we do not lose any generality by restricting our discussion to quadratic forms that are constructed from symmetric matrices.

Victor Salazar
Victor Salazar
Numerade Educator
04:37

Problem 15

There is an alternative criterion for positive definiteness based on subdeterminants of the matrix. The $2 \times 2$ version already appears in (3.70). (a) Prove that a $3 \times 3$ matrix $K=\left(\begin{array}{lll}a & b & c \\ b & d & e \\ c & e & f\end{array}\right)$ is positive definite if and only if $a>0, a d-b^2>0$, and $\operatorname{det} K>0$.
(b) Prove the general version: an $n \times n$ matrix $K>0$ is positive definite if and only if its upper left square $k \times k$ submatrices have positive determinant for all $k=1, \ldots, n$.

Runpeng Li
Runpeng Li
Numerade Educator
01:10

Problem 15

Prove that $e^{z+w}=e^z e^w$. Conclude that $e^{m z}=\left(e^z\right)^m$ whenever $m$ is an integer.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:07

Problem 16

Prove that $A=A^T$ is a symmetric $n \times n$ matrix if and only if $(A \mathbf{v}) \cdot \mathbf{w}=\mathbf{v} \cdot(A \mathbf{w})$ for all $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$.

Sikandar Baig
Sikandar Baig
Numerade Educator
03:26

Problem 16

Find all vectors in $\mathbb{R}^3$ that are orthogonal to both $(1,2,3)^T$ and $(-2,0,1)^T$.

Victor Salazar
Victor Salazar
Numerade Educator
07:46

Problem 16

Prove that $\lim _{p \rightarrow \infty}\|\mathbf{v}\|_p=\|\mathbf{v}\|_{\infty}$ for all $\mathbf{v} \in \mathbb{R}^2$.

Shafiq Rehman
Shafiq Rehman
Numerade Educator
View

Problem 16

(a) Show that a symmetric matrix $N$ is negative definite if and only if $K=-N$ is positive definite. (b) Write down two explicit criteria that tell whether a $2 \times 2$ matrix $N=\left(\begin{array}{ll}a & b \\ b & c\end{array}\right)$ is negative definite.
(c) Use your criteria to check whether
(i) $\left(\begin{array}{rr}-1 & 1 \\ 1 & -2\end{array}\right)$,
(ii) $\left(\begin{array}{ll}-4 & -5 \\ -5 & -6\end{array}\right)$,
(iii) $\left(\begin{array}{rr}-3 & -1 \\ -1 & 2\end{array}\right)$ are negative definite.

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 16

Let $K$ be a symmetric matrix. Prove that if a non-positive diagonal entry appears anywhere (not necessarily in the pivot position) in the matrix during Regular Gaussian Elimination, then $K$ is not positive definite.

Victor Salazar
Victor Salazar
Numerade Educator
01:09

Problem 16

(a) Use the formula $e^{2 \mathrm{i} \theta}=\left(e^{\mathrm{i} \theta}\right)^2$ to deduce the well-known trigonometric identities for $\cos 2 \theta$ and $\sin 2 \theta$. (b) Derive the corresponding identities for $\cos 3 \theta$ and $\sin 3 \theta$.
(c) Write down the explicit identities for $\cos m \theta$ and $\sin m \theta$ as polynomials in $\cos \theta$ and $\sin \theta$.

Chai Santi
Chai Santi
Numerade Educator
06:33

Problem 17

Prove that $\langle A, B\rangle=\operatorname{tr}\left(A^T B\right)$ defines an inner product on the vector space $\mathcal{M}_{n \times n}$ of real $n \times n$ matrices.

Chris Trentman
Chris Trentman
Numerade Educator
00:46

Problem 17

Answer Exercises 3.2 .15 and 3.2 .16 using the weighted inner product
$$
\langle\mathbf{v}, \mathbf{w}\rangle=3 v_1 w_1+2 v_2 w_2+v_3 w_3 \text {. }
$$

Monica Miller
Monica Miller
Numerade Educator
13:31

Problem 17

Justify the triangle inequality for (a) the $\mathrm{L}^1$ norm (3.31); (b) the $\mathrm{L}^{\infty}$ norm (3.32).

Chris Trentman
Chris Trentman
Numerade Educator
View

Problem 17

Show that $\mathbf{x}=\left(\begin{array}{l}1 \\ 1\end{array}\right)$ is a null direction for $K=\left(\begin{array}{rr}1 & -2 \\ -2 & 3\end{array}\right)$, but $\mathbf{x} \notin \operatorname{ker} K$.

Nick Johnson
Nick Johnson
Numerade Educator
01:00

Problem 17

Formulate a determinantal criterion similar to that in Exercise 3.5 .15 for negative definite matrices. Write out the $2 \times 2$ and $3 \times 3$ cases explicitly.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:42

Problem 17

Use complex exponentials to prove the identity $\cos \theta-\cos \varphi=2 \cos \frac{\theta-\varphi}{2} \cos \frac{\theta+\varphi}{2}$.

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
06:33

Problem 18

Suppose $\langle\mathbf{v}, \mathbf{w}\rangle$ defines an inner product on a vector space $V$. Explain why it also defines an inner product on every subspace $W \subset V$.

Chris Trentman
Chris Trentman
Numerade Educator
18:14

Problem 18

Find all vectors in $\mathbb{R}^4$ that are orthogonal to both $(1,2,3,4)^T$ and $(5,6,7,8)^T$.

Donald Albin
Donald Albin
Numerade Educator
02:19

Problem 18

Let $w(x)>0$ for $a \leq x \leq b$ be a weight function. (a) Prove that
$\|f\|_{1, w}=\int_a^b|f(x)| w(x) d x$ defines a norm on $\mathrm{C}^0[a, b]$, called the weighted $\mathrm{L}^1$ norm.
(b) Do the same for the weighted $\mathrm{L}^{\infty}$ norm $\|f\|_{\infty, w}=\max \{|f(x)| w(x): a \leq x \leq b\}$.

Regina Hays
Regina Hays
Numerade Educator
01:06

Problem 18

Explain why an indefinite quadratic form necessarily has a non-zero null direction.

Christine Girgus
Christine Girgus
Numerade Educator
01:30

Problem 18

True or false: A negative definite matrix must have negative trace and negative determinant.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:27

Problem 18

Prove that if $z=x+\mathrm{i} y$, then $\left|e^z\right|=e^x, \quad \operatorname{ph} e^z=y$.

Sandip Ranjan
Sandip Ranjan
Numerade Educator
03:33

Problem 19

Prove that if $\langle\mathbf{v}, \mathbf{w}\rangle$ and $\langle\langle\mathbf{v}, \mathbf{w}\rangle\rangle$ are two inner products on the same vector space $V$, then their sum $\langle\langle\langle\mathbf{v}, \mathbf{w}\rangle\rangle\rangle=\langle\mathbf{v}, \mathbf{w}\rangle+\langle\langle\mathbf{v}, \mathbf{w}\rangle\rangle$ defines an inner product on $V$.

Donald Albin
Donald Albin
Numerade Educator
05:24

Problem 19

Determine a basis for the subspace $W \subset \mathbb{R}^4$ consisting of all vectors which are orthogonal to the vector $(1,2,-1,3)^T$.

Derrick Danso
Derrick Danso
Numerade Educator
13:31

Problem 19

Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two different norms on a vector space $V$. (a) Prove that $\|\mathbf{v}\|=\max \left\{\|\mathbf{v}\|_1,\|\mathbf{v}\|_2\right\}$ defines a norm on $V$. (b) Does $\|\mathbf{v}\|=\min \left\{\|\mathbf{v}\|_1,\|\mathbf{v}\|_2\right\}$ define a norm? (c) Does the arithmetic mean $\|\mathbf{v}\|=\frac{1}{2}\left(\|\mathbf{v}\|_1+\|\mathbf{v}\|_2\right)$ define a norm?
(d) Does the geometric mean $\|\mathbf{v}\|=\sqrt{\|\mathbf{v}\|_1\|\mathbf{v}\|_2}$ define a norm?

Chris Trentman
Chris Trentman
Numerade Educator
01:05

Problem 19

Let $K=K^T$. True or false: (a) If $K$ admits a null direction, then $\operatorname{ker} K \neq\{0\}$.
(b) If $K$ has no null directions, then $K$ is either positive or negative definite.

Linh Vu
Linh Vu
Numerade Educator

Problem 19

Find the Cholesky factorizations of the following matrices:
(a)
$$
\begin{aligned}
& \left(\begin{array}{rr}
3 & -2 \\
-2 & 2
\end{array}\right), \\
& \text { (e) }\left(\begin{array}{llll}
2 & 1 & 0 & 0 \\
1 & 2 & 1 & 0 \\
0 & 1 & 2 & 1 \\
0 & 0 & 1 & 2
\end{array}\right) .
\end{aligned}
$$
(b) $\left(\begin{array}{rr}4 & -12 \\ -12 & 45\end{array}\right)$,
(c) $\left(\begin{array}{rrr}1 & 1 & 1 \\ 1 & 2 & -2 \\ 1 & -2 & 14\end{array}\right)$,
(d) $\left(\begin{array}{lll}2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2\end{array}\right)$,
(e) $\left(\begin{array}{llll}2 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 2\end{array}\right)$.

Check back soon!
01:05

Problem 19

The formulas $\cos z=\frac{e^{\mathrm{i} z}+e^{-\mathrm{i} z}}{2}$ and $\sin z=\frac{e^{\mathrm{i} z}-e^{-\mathrm{i} z}}{2 \mathrm{i}}$ serve to define the basic complex trigonometric functions. Write out the formulas for their real and imaginary parts in terms of $z=x+\mathrm{i} y$, and show that $\cos z$ and $\sin z$ reduce to their usual real forms when $z=x$ is real. What do they become when $z=\mathrm{i} y$ is purely imaginary?

Aman Gupta
Aman Gupta
Numerade Educator

Problem 20

Let $V$ and $W$ be inner product spaces with respective inner products $\langle\mathbf{v}, \overrightarrow{\mathbf{v}}\rangle$ and $\langle\langle\mathbf{w}, \overline{\mathbf{w}}\rangle\rangle$. Show that $\langle\langle\langle(\mathbf{v}, \mathbf{w}),(\overline{\mathbf{v}}, \overline{\mathbf{w}})\rangle\rangle\rangle=\langle\mathbf{v}, \overline{\mathbf{v}}\rangle+\langle\langle\mathbf{w}, \overline{\mathbf{w}}\rangle\rangle$ for $\mathbf{v}, \overline{\mathbf{v}} \in V, \mathbf{w}, \overline{\mathbf{w}} \in W$, defines an inner product on their Cartesian product $V \times W$.

Check back soon!
01:34

Problem 20

Find three vectors $\mathbf{u}, \mathbf{v}$ and $\mathbf{w}$ in $\mathbb{R}^3$ such that $\mathbf{u}$ and $\mathbf{v}$ are orthogonal, $\mathbf{u}$ and $\mathbf{w}$ are orthogonal, but $\mathbf{v}$ and $\mathbf{w}$ are not orthogonal. Are your vectors linearly independent or linearly dependent? Can you find vectors of the opposite dependency satisfying the same conditions? Why or why not?

Linh Vu
Linh Vu
Numerade Educator
02:41

Problem 20

Find a unit vector in the same direction as $\mathbf{v}=(1,2,-3)^T$ for (a) the Euclidean norm,
(b) the weighted norm $\|\mathbf{v}\|^2=2 v_1^2+v_2^2+\frac{1}{3} v_3^2$, (c) the 1 norm, (d) the $\infty$ norm, (e) the norm based on the inner product $2 v_1 w_1-v_1 w_2-v_2 w_1+2 v_2 w_2-v_2 w_3-v_3 w_2+2 v_3 w_3$.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:25

Problem 20

In special relativity, light rays in Minkowski space-time $\mathbb{R}^n$ travel along the light cone which, by definition, consists of all null directions associated with an indefinite quadratic form $q(\mathrm{x})=\mathrm{x}^T K \mathrm{x}$. Find and sketch a picture of the light cone when the coefficient matrix
$K$ is (a) $\left(\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right)$,
(b) $\left(\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right)$,
(c) $\left(\begin{array}{rrr}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{array}\right)$. Remark. In the physical universe, space-time is $n=4$-dimensional, and $K$ is given in (3.57), [55].

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 20

Which of the matrices in Exercise 3.5.1 have a Cholesky factorization? For those that do, write out the factorization.

Victor Salazar
Victor Salazar
Numerade Educator
04:56

Problem 20

The complex hyperbolic functions are defined as $\cosh z=\frac{e^z+e^{-z}}{2}, \sinh z=\frac{e^z-e^{-z}}{2}$
(a) Write out the formulas for their real and imaginary parts in terms of $z=x+\mathrm{i} y$.
(b) Prove that $\cos \mathrm{i} z=\cosh z$ and $\sin \mathrm{i} z=\mathrm{i} \sinh z$.

Suzanne W.
Suzanne W.
Numerade Educator
04:23

Problem 21

For each of the given pairs of functions in $\mathrm{C}^0[0,1]$, find their $\mathrm{L}^2$ inner product $\langle f, g\rangle$ and their $\mathrm{L}^2$ norms $\|f\|,\|g\|:$ (a) $f(x)=1, g(x)=x ;$ (b) $f(x)=\cos 2 \pi x$, $g(x)=\sin 2 \pi x ;$
(c) $f(x)=x, g(x)=e^x$;
(d) $f(x)=(x+1)^2, g(x)=\frac{1}{x+1}$.

Caleb Wood
Caleb Wood
Numerade Educator
07:12

Problem 21

For what values of $a, b$ are the vectors $(1,1, a)^T$ and $(b,-1,1)^T$ orthogonal
(a) with respect to the dot product?
(b) with respect to the weighted inner product of Exercise 3.2.17?

Anthony Ramos
Anthony Ramos
Numerade Educator
03:01

Problem 21

Show that, for every choice of given angles $\theta, \phi$, and $\psi$, the following are unit vectors in the Euclidean norm: (a) $(\cos \theta \cos \phi, \cos \theta \sin \phi, \sin \theta)^T$.
(c) $(\cos \theta \cos \phi \cos \psi, \cos \theta \cos \phi \sin \psi, \cos \theta \sin \phi, \sin \theta)^T$.
(b) $\frac{1}{\sqrt{2}}(\cos \theta, \sin \theta, \cos \phi, \sin \phi)^T$.

AG
Ankit Gupta
Numerade Educator
02:57

Problem 21

A function $f(\mathbf{x})$ on $\mathbb{R}^n$ is called homogeneous of degree $k$ if $f(c \mathbf{x})=c^k f(\mathbf{x})$ for all scalars c. (a) Given $\mathbf{a} \in \mathbb{R}^n$, show that the linear form $\ell(\mathbf{x})=\mathbf{a} \cdot \mathbf{x}=a_1 x_1+\cdots+a_n x_n$ is homogeneous of degree 1 . (b) Show that the quadratic form $q(\mathbf{x})=\mathbf{x}^T K \mathbf{x}=\sum_{i, j=1}^n k_{i j} x_i x_j$ is homogeneous of degree 2.
(c) Find a homogeneous function of degree 2 on $\mathbb{R}^2$ that is not a quadratic form.

Adrian Co
Adrian Co
Numerade Educator
06:37

Problem 21

Write the following positive definite quadratic forms as a sum of pure squares, as in (3.79):
(a) $16 x_1^2+25 x_2^2$,
(b) $x_1^2-2 x_1 x_2+4 x_2^2$,
(c) $5 x_1^2+4 x_1 x_2+3 x_2^2$,
(d) $3 x_1^2-2 x_1 x_2-2 x_1 x_3+2 x_2^2+6 x_3^2$,
(e) $x_1^2+x_1 x_2+x_2^2+x_2 x_3+x_3^2$,
(f) $4 x_1^2-2 x_1 x_2-4 x_1 x_3+\frac{1}{2} x_2^2-x_2 x_3+6 x_3^2$,
(g) $3 x_1^2+2 x_1 x_2+3 x_2^2+2 x_2 x_3+3 x_3^2+2 x_3 x_4+3 x_4^2$.

Victor Salazar
Victor Salazar
Numerade Educator
02:22

Problem 21

Generalizing Example 2.17c, by a trigonometric polynomial of degree $\leq n$, we mean a function $T(x)=\sum_{0 \leq j+k \leq n} c_{j k}(\cos \theta)^j(\sin \theta)^k$ in the powers of the sine and cosine functions up to degree $n$. (a) Use formula (3.94) to prove that every trigonometric polynomial of degree $\leq n$ can be written as a complex linear combination of the $2 n+1$ complex exponentials $e^{-n \mathrm{i} \theta}, \ldots e^{-\mathrm{i} \theta}, e^{0 \mathrm{i} \theta}=1, e^{\mathrm{i} \theta}, e^{2 \mathrm{i} \theta}, \ldots e^{n \mathrm{i} \theta}$. (b) Prove that every trigonometric polynomial of degree $\leq n$ can be written as a real linear combination of the trigonometric functions $1, \cos \theta, \sin \theta, \cos 2 \theta, \sin 2 \theta, \ldots \cos n \theta, \sin n \theta$.
(c) Write out the following trigonometric polynomials in both of the preceding forms:
(i) $\cos ^2 \theta$, (ii) $\cos \theta \sin \theta$, (iii) $\cos ^3 \theta$, (iv) $\sin ^4 \theta$, (v) $\cos ^2 \theta \sin ^2 \theta$.

AG
Ankit Gupta
Numerade Educator
07:04

Problem 22

Let $f(x)=x, g(x)=1+x^2$. Compute $\langle f, g\rangle,\|f\|$, and $\|g\|$ for (a) the $\mathrm{L}^2$ inner product $\langle f, g\rangle=\int_0^1 f(x) g(x) d x ;(b)$ the $\mathrm{L}^2$ inner product $\langle f, g\rangle=\int_{-1}^1 f(x) g(x) d x$; (c) the weighted inner product $\langle f, g\rangle=\int_0^1 f(x) g(x) x d x$.

Ahmad Reda
Ahmad Reda
Numerade Educator
00:45

Problem 22

When is a vector orthogonal to itself?

AG
Ankit Gupta
Numerade Educator
01:08

Problem 22

How many unit vectors are parallel to a given vector $\mathbf{v} \neq \mathbf{0}$ ? (a) 1, (b) 2 , (c) 3 ,
(d) $\infty,(e)$ depends on the norm. Explain your answer.

Ankur S
Ankur S
Numerade Educator
01:17

Problem 22

(a) Find the Gram matrix corresponding to each of the following sets of vectors using the Euclidean dot product on $\mathbb{R}^n$.
(i) $\left(\begin{array}{r}-1 \\ 3\end{array}\right),\left(\begin{array}{l}0 \\ 2\end{array}\right)$,
(ii) $\left(\begin{array}{l}1 \\ 2\end{array}\right),\left(\begin{array}{r}-2 \\ 3\end{array}\right),\left(\begin{array}{l}-1 \\ -1\end{array}\right)$,
(iii) $\left(\begin{array}{r}2 \\ 1 \\ -1\end{array}\right),\left(\begin{array}{r}-3 \\ 0 \\ 2\end{array}\right)$,
(iv) $\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right),\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)$,
(v) $\left(\begin{array}{r}1 \\ -2 \\ 2\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ -1 \\ 1\end{array}\right)$,
(vi) $\left(\begin{array}{r}1 \\ 0 \\ -1 \\ 0\end{array}\right),\left(\begin{array}{r}-1 \\ 1 \\ 0 \\ 1\end{array}\right)$,
(vii)
$\left(\begin{array}{l}1 \\ 2 \\ 3 \\ 4\end{array}\right),\left(\begin{array}{r}-2 \\ 1 \\ -4 \\ 3\end{array}\right),\left(\begin{array}{r}-1 \\ 3 \\ -1 \\ -2\end{array}\right)$,
(viii)
$\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{r}-2 \\ 1 \\ 0 \\ 0\end{array}\right),\left(\begin{array}{r}-1 \\ 0 \\ -1 \\ 0\end{array}\right),\left(\begin{array}{r}0 \\ 2 \\ -3 \\ 0\end{array}\right)$.
(b) Which are positive definite? (c) If the matrix is positive semi-definite, find all its null directions.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:45

Problem 22

Write out the real and imaginary parts of the power function $x^c$ with complex exponent $c=a+\mathrm{i} b \in \mathbb{C}$.

Willis James
Willis James
Numerade Educator
08:44

Problem 23

Which of the following formulas for $\langle f, g\rangle$ define inner products on the space $\mathrm{C}^0[-1,1]$ ?
(a) $\int_{-1}^1 f(x) g(x) e^{-x} d x$
(b) $\int_{-1}^1 f(x) g(x) x d x$,
(c) $\int_{-1}^1 f(x) g(x)(x+2) d x$,
(d) $\int_{-1}^1 f(x) g(x) x^2 d x$.

Ahmad Reda
Ahmad Reda
Numerade Educator
05:30

Problem 23

Prove that the only element $\mathbf{w}$ in an inner product space $V$ that is orthogonal to every vector, so $\langle\mathbf{w}, \mathbf{v}\rangle=0$ for all $\mathbf{v} \in V$, is the zero vector: $\mathbf{w}=\mathbf{0}$.

Anthony Ramos
Anthony Ramos
Numerade Educator
04:46

Problem 23

Plot the unit circle (sphere) for (a) the weighted norm $\|\mathbf{v}\|=\sqrt{v_1^2+4 v_2^2}$;
(b) the norm based on the inner product (3.9); (c) the norm of Exercise 3.3.9.

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
View

Problem 23

Recompute the Gram matrices for cases (iii) $-(v)$ in the previous exercise using the weighted inner product $\langle\mathbf{x}, \mathbf{y}\rangle=x_1 y_1+2 x_2 y_2+3 x_3 y_3$. Does this change its positive definiteness?

Victor Salazar
Victor Salazar
Numerade Educator
04:59

Problem 23

Write the power series expansions for $e^{\mathrm{i} x}$. Prove that the real terms give the power series for $\cos x$, while the imaginary terms give that of $\sin x$. Use this identification to justify Euler's formula (3.92).

Khaled Yasein
Khaled Yasein
Numerade Educator
03:23

Problem 24

Prove that $\langle f, g\rangle=\int_0^1 f(x) g(x) d x$ does not define an inner product on the vector space $\mathrm{C}^0[-1,1]$. Explain why this does not contradict the fact that it defines an inner product on the vector space $\mathrm{C}^0[0,1]$. Does it define an inner product on the subspace $\mathcal{P}^{(n)} \subset \mathrm{C}^0[-1,1]$ consisting of all polynomial functions?

Donald Albin
Donald Albin
Numerade Educator
00:59

Problem 24

A vector with $\|\mathbf{v}\|=1$ is known as a unit vector. Prove that if $\mathbf{v}, \mathbf{w}$ are both unit vectors, then $\mathbf{v}+\mathbf{w}$ and $\mathbf{v}-\mathbf{w}$ are orthogonal. Are they also unit vectors?

James Kiss
James Kiss
Numerade Educator
01:22

Problem 24

Draw the unit circle for each norm in Exercise 3.3.10.

Abhijith V
Abhijith V
Numerade Educator
00:26

Problem 24

Recompute the Gram matrices for cases (vi)-(viii) in Exercise 3.4.22 for the weighted inner product $\langle\mathbf{x}, \mathbf{y}\rangle=x_1 y_1+\frac{1}{2} x_2 y_2+\frac{1}{3} x_3 y_3+\frac{1}{4} x_4 y_4$.

Wendi Zhao
Wendi Zhao
Numerade Educator
04:34

Problem 24

The derivative of a complex-valued function $f(x)=u(x)+\mathrm{i} v(x)$, depending on a real variable $x$, is given by $f^{\prime}(x)=u^{\prime}(x)+\mathrm{i} v^{\prime}(x)$. (a) Prove that if $\lambda=\mu+\mathrm{i} \nu$ is any complex scalar, then $\frac{d}{d x} e^{\lambda x}=\lambda e^{\lambda x}$. (b) Prove, conversely, $\int_a^b e^{\lambda x} d x=\frac{1}{\lambda}\left(e^{\lambda b}-e^{\lambda a}\right)$ provided $\lambda \neq 0$.

Chris Trentman
Chris Trentman
Numerade Educator
03:23

Problem 25

Does either of the following define an inner product on $\mathrm{C}^0[0,1]$ ?
(a) $\langle f, g\rangle=f(0) g(0)+f(1) g(1)$,
(b) $\langle f, g\rangle=f(0) g(0)+f(1) g(1)+\int_0^1 f(x) g(x) d x$.

Donald Albin
Donald Albin
Numerade Educator
05:30

Problem 25

Let $V$ be an inner product space and $\mathbf{v} \in V$. Prove that the set of all vectors $\mathbf{w} \in V$ that are orthogonal to $\mathbf{v}$ is a subspace of $V$.

Anthony Ramos
Anthony Ramos
Numerade Educator

Problem 25

Sketch the unit sphere $S_1 \subset \mathbb{R}^3$ for
(a) the $\mathrm{L}^1$ norm, (b) the $\mathrm{L}^{\infty}$ norm, (c) the weighted norm $\|\mathbf{v}\|^2=2 v_1^2+v_2^2+3 v_3^2$,
(d) $\|\mathbf{v}\|=\max \left\{\left|v_1+v_2\right|,\left|v_1+v_3\right|,\left|v_2+v_3\right|\right\}$.

Check back soon!

Problem 25

Find the Gram matrix $K$ for the functions $1, e^x, e^{2 x}$ using the $\mathrm{L}^2$ inner product on $[0,1]$. Is $K$ positive definite?

Check back soon!
01:23

Problem 25

Use the complex trigonometric formulas (3.94) and Exercise 3.6.24 to evaluate the following trigonometric integrals:
(a) $\int \cos ^2 x d x$
(b) $\int \sin ^2 x d x$,
(c) $\int \cos x \sin x d x$
(d) $\int \cos 3 x \sin 5 x d x$. How did you calculate them in first-year calculus? If you're not convinced this method is easier, try the more complicated integrals
(e) $\int \cos ^4 x d x$,
(f) $\int \sin ^4 x d x$,
(g) $\int \cos ^2 x \sin ^2 x d x$,
(h) $\int \cos 3 x \sin 5 x \cos 7 x d x$.

Gregory Higby
Gregory Higby
Numerade Educator
02:57

Problem 26

Let $f(x)$ be a function, and $\|f\|$ its $\mathrm{L}^2$ norm on $[a, b]$. Is $\left\|f^2\right\|=\|f\|^2$ ? If yes, prove the statement. If no, give a counterexample.

Vysakh M
Vysakh M
Numerade Educator
06:12

Problem 26

(a) Show that the polynomials $p_1(x)=1, p_2(x)=x-\frac{1}{2}, p_3(x)=x^2-x+\frac{1}{6}$ are mutually orthogonal with respect to the $\mathrm{L}^2$ inner product on the interval $[0,1]$.
(b) Show that the functions $\sin n \pi x, n=1,2,3, \ldots$, are mutually orthogonal with respect to the same inner product.

Donald Albin
Donald Albin
Numerade Educator

Problem 26

Let $\mathbf{v} \neq \mathbf{0}$ be any nonzero vector in a normed vector space $V$. Show how to construct a new norm on $V$ that changes $\mathbf{v}$ into a unit vector.

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01:13

Problem 26

Answer Exercise 3.4.25 using the weighted inner product $\langle f, g\rangle=\int_0^1 f(x) g(x) e^{-x} d x$.

Wendi Zhao
Wendi Zhao
Numerade Educator
00:50

Problem 26

Determine whether the indicated sets of complex vectors are linearly independent or dependent.
(a) $\left(\begin{array}{l}\mathrm{i} \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ \mathrm{i}\end{array}\right)$,
(b) $\left(\begin{array}{c}1+\mathrm{i} \\ 1\end{array}\right),\left(\begin{array}{c}2 \\ 1-\mathrm{i}\end{array}\right)$,
(c) $\left(\begin{array}{c}1+3 \mathrm{i} \\ 2-\mathrm{i}\end{array}\right),\left(\begin{array}{c}2-3 \mathrm{i} \\ 1-\mathrm{i}\end{array}\right)$,
(d) $\left(\begin{array}{c}-2+\mathrm{i} \\ \mathrm{i}\end{array}\right),\left(\begin{array}{c}4-3 \mathrm{i} \\ 1\end{array}\right),\left(\begin{array}{c}2 \mathrm{i} \\ 1-5 \mathrm{i}\end{array}\right)$,
(e) $\left(\begin{array}{c}1+2 \mathrm{i} \\ 2 \\ 0\end{array}\right),\left(\begin{array}{c}2 \\ 0 \\ 1-\mathrm{i}\end{array}\right)$,
(f) $\left(\begin{array}{c}1 \\ 3 \mathrm{i} \\ 2-\mathrm{i}\end{array}\right),\left(\begin{array}{c}1+2 \mathrm{i} \\ -3 \\ 0\end{array}\right),\left(\begin{array}{c}1-\mathrm{i} \\ -\mathrm{i} \\ 1\end{array}\right)$,
(g) $\left(\begin{array}{c}1+\mathrm{i} \\ 2-\mathrm{i} \\ 1\end{array}\right),\left(\begin{array}{c}1-\mathrm{i} \\ -3 \mathrm{i} \\ 1-2 \mathrm{i}\end{array}\right),\left(\begin{array}{c}-1+\mathrm{i} \\ 2+3 \mathrm{i} \\ 1+2 \mathrm{i}\end{array}\right)$.

Donald Albin
Donald Albin
Numerade Educator
06:27

Problem 27

Prove that $\langle f, g\rangle=\int_a^b\left[f(x) g(x)+f^{\prime}(x) g^{\prime}(x)\right] d x$ defines an inner product on the space $\mathrm{C}^1[a, b]$ of continuously differentiable functions on the interval $[a, b]$. Write out the corresponding norm, known as the Sobolev $\mathrm{H}^1$ norm; it and its generalizations play an extremely important role in advanced mathematical analysis, [49].

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
02:25

Problem 27

Find a non-zero quadratic polynomial that is orthogonal to both $p_1(x)=1$ and $p_2(x)=x$ under the $\mathrm{L}^2$ inner product on the interval $[-1,1]$.

Jimmy Yao
Jimmy Yao
Numerade Educator
01:14

Problem 27

True or false: Two norms on a vector space have the same unit sphere if and only if they are the same norm.

Linh Vu
Linh Vu
Numerade Educator
01:40

Problem 27

Find the Gram matrix $K$ for the monomials $1, x, x^2, x^3$ using the $\mathrm{L}^2$ inner product on $[-1,1]$. Is $K$ positive definite?

AG
Ankit Gupta
Numerade Educator
00:38

Problem 27

True or false: The set of complex vectors of the form $\left(\frac{z}{z}\right)$ for $z \in \mathbb{C}$ is a subspace of $\mathbb{C}^2$.

Hossam Mohamed
Hossam Mohamed
Numerade Educator
08:42

Problem 28

Let $V=\mathrm{C}^1[-1,1]$ denote the vector space of continuously differentiable functions for $-1 \leq x \leq 1$. (a) Does the expression $\langle f, g\rangle=\int_{-1}^1 f^{\prime}(x) g^{\prime}(x) d x$ define an inner product on $V$ ? (b) Answer the same question for the subspace $W=\{f \in V \mid f(0)=0\}$ consisting of all continuously differentiable functions that vanish at 0 .

Patrick Vaughn
Patrick Vaughn
Numerade Educator
01:10

Problem 28

Find all quadratic polynomials that are orthogonal to the function $e^x$ with respect to the $\mathrm{L}^2$ inner product on the interval $[0,1]$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
08:24

Problem 28

Find the unit function that is a constant multiple of the function $f(x)=x-\frac{1}{3}$ with respect to the (a) $\mathrm{L}^1$ norm on $[0,1] ;(b) \mathrm{L}^2$ norm on $[0,1] ;(c) \mathrm{L}^{\infty}$ norm on $[0,1] ;(d) \mathrm{L}^1$ norm on $[-1,1] ;(e) L^2$ norm on $[-1,1] ;(f) L^{\infty}$ norm on $[-1,1]$.

Matthew Allcock
Matthew Allcock
Numerade Educator
08:44

Problem 28

Answer Exercise 3.4.27 using the weighted inner product $\langle f, g\rangle=\int_{-1}^1 f(x) g(x)(1+x) d x$

Ahmad Reda
Ahmad Reda
Numerade Educator

Problem 28

(a) Determine whether the vectors $\mathbf{v}_1=\left(\begin{array}{l}1 \\ \mathrm{i} \\ 0\end{array}\right), \mathbf{v}_2=\left(\begin{array}{c}0 \\ 1+\mathrm{i} \\ 2\end{array}\right), \mathbf{v}_3=\left(\begin{array}{c}-1+\mathrm{i} \\ 1+\mathrm{i} \\ -1\end{array}\right)$, are linearly independent or linearly dependent. (b) Do they form a basis of $\mathbb{C}^3$ ? (c) Compute the Hermitian norm of each vector. (d) Compute the Hermitian dot products between all different pairs. Which vectors are orthogonal?

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04:10

Problem 29

(a) Let $h(x) \geq 0$ be a continuous, non-negative function defined on an interval $[a, b]$. Prove that $\int_a^b h(x) d x=0$ if and only if $h(x) \equiv 0$. (b) Give an example that shows that this result is not valid if $h$ is allowed to be discontinuous.

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator
02:57

Problem 29

Determine all pairs among the functions $1, x, \cos \pi x, \sin \pi x, e^x$, that are orthogonal with respect to the $\mathrm{L}^2$ inner product on $[-1,1]$.
3.2.30. Find two non-zero functions that are orthogonal with respect to the weighted inner product $\langle f, g\rangle=\int_0^1 f(x) g(x) x d x$.

Fuzail Shakir
Fuzail Shakir
Numerade Educator
08:24

Problem 29

For which norms is the constant function $f(x) \equiv 1$ a unit function?
(a) $\mathrm{L}^1$ norm on $[0,1]$; (b) $\mathrm{L}^2$ norm on $[0,1]$; (c) $\mathrm{L}^{\infty}$ norm on $[0,1]$;
(d) $\mathrm{L}^1$ norm on $[-1,1]$; (e) $\mathrm{L}^2$ norm on $[-1,1]$; (f) $\mathrm{L}^{\infty}$ norm on $[-1,1]$;
(g) $\mathrm{L}^1$ norm on $\mathbb{R}$; $(h) \mathrm{L}^2$ norm on $\mathbb{R}$; (i) $\mathrm{L}^{\infty}$ norm on $\mathbb{R}$.

Matthew Allcock
Matthew Allcock
Numerade Educator
01:22

Problem 29

Let $K$ be a $2 \times 2$ Gram matrix. Explain why the positive definiteness criterion (3.55) is equivalent to the Cauchy-Schwarz inequality.

Rukhmani Jain
Rukhmani Jain
Numerade Educator
03:57

Problem 29

Find the dimension of and a basis for the following subspaces of $\mathbb{C}^3$ : (a) The set of all complex multiples of $(1, \mathrm{i}, 1-\mathrm{i})^T$. (b) The plane $z_1+\mathrm{i} z_2+(1-\mathrm{i}) z_3=0$. (c) The image of the matrix $A=\left(\begin{array}{ccc}1 & \mathrm{i} & 2-\mathrm{i} \\ 2+\mathrm{i} & 1+3 \mathrm{i} & -1-\mathrm{i}\end{array}\right)$.
(d) The kernel of the same matrix. (e) The set of vectors that are orthogonal to $(1-\mathrm{i}, 2 \mathrm{i}, 1+\mathrm{i})^T$.

Cory Glover
Cory Glover
Numerade Educator
02:20

Problem 30

(a) Prove the inner product axioms for the weighted inner product (3.15), assuming $w(x)>0$ for all $a \leq x \leq b$. (b) Explain why it does not define an inner product if $w$ is continuous and $w\left(x_0\right)<0$ for some $x_0 \in[a, b]$. (c) If $w(x) \geq 0$ for $a \leq x \leq b$, does (3.15) define an inner product?

Donald Albin
Donald Albin
Numerade Educator

Problem 30

A subset $S \subset \mathbb{R}^n$ is called convex if, for all $\mathbf{x}, \mathbf{y} \in S$, the line segment joining $\mathbf{x}$ to $\mathbf{y}$ is also in $S$, i.e., $t \mathbf{x}+(1-t) \mathbf{y} \in S$ for all $0 \leq t \leq 1$. Prove that the unit ball is a convex subset of a normed vector space. Is the unit sphere convex?

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03:20

Problem 30

(a) Prove that if $K$ is a positive definite matrix, then $K^2$ is also positive definite.
(b) More generally, if $S=S^T$ is symmetric and nonsingular, then $S^2$ is positive definite.

Chris Trentman
Chris Trentman
Numerade Educator
14:34

Problem 30

Find bases for the four fundamental subspaces associated with the complex matrices
(a) $\left(\begin{array}{cc}\mathrm{i} & 2 \\ -1 & 2 \mathrm{i}\end{array}\right)$,
(b) $\left(\begin{array}{ccc}2 & -1+\mathrm{i} & 1-2 \mathrm{i} \\ -4 & 3-\mathrm{i} & 1+\mathrm{i}\end{array}\right)$,
(c) $\left(\begin{array}{ccc}\mathrm{i} & -1 & 2-\mathrm{i} \\ -1+2 \mathrm{i} & -2-\mathrm{i} & 3 \\ \mathrm{i} & -1 & 1+\mathrm{i}\end{array}\right)$.

Uma Kumari
Uma Kumari
Numerade Educator

Problem 31

Let $\Omega \subset \mathbb{R}^2$ be a closed bounded subset. Let $C^0(\Omega)$ denote the vector space consisting of all continuous, bounded real-valued functions $f(x, y)$ defined for $(x, y) \in \Omega$. (a) Prove that if $f(x, y) \geq 0$ is continuous and $\iint_{\Omega} f(x, y) d x d y=0$, then $f(x, y) \equiv 0$. (b) Use this result to prove that
$$
\langle f, g\rangle=\iint_{\Omega} f(x, y) g(x, y) d x d y
$$
defines an inner product on $\mathrm{C}^0(\Omega)$, called the $\mathrm{L}^2$ inner product on the domain $\Omega$. What is the corresponding norm?

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00:50

Problem 31

Use the dot product on $\mathbb{R}^3$ to answer the following: (a) Find the angle between the vectors $(1,2,3)^T$ and $(1,-1,2)^T$. (b) Verify the Cauchy-Schwarz and triangle inequalities for these two particular vectors. (c) Find all vectors that are orthogonal to both of these vectors.

Jodi Folley
Jodi Folley
Numerade Educator
View

Problem 31

Check the validity of the inequalities (3.38) for the particular vectors
(a) $(1,-1)^T$,
(b) $(1,2,3)^T$,
(c) $(1,1,1,1)^T$,
(d) $(1,-1,-2,-1,1)^T$.

Sarah Parrigin
Sarah Parrigin
Numerade Educator
02:25

Problem 31

Let $A$ be an $m \times n$ matrix. (a) Explain why the product $L=A A^T$ is a Gram matrix.
(b) Show that, even though they may be of different sizes, both Gram matrices $K=A^T A$ and $L=A A^T$ have the same rank. (c) Under what conditions are both $K$ and $L$ positive definite?

Nick Johnson
Nick Johnson
Numerade Educator
08:21

Problem 31

Prove that $\mathbf{v}=\mathbf{x}+\mathrm{i} \mathbf{y}$ and $\mathbf{\nabla}=\mathbf{x}-\mathrm{i} \mathbf{y}$ are linearly independent complex vectors if and only if their real and imaginary parts $\mathbf{x}$ and $\mathbf{y}$ are linearly independent real vectors.

Brian Beasley
Brian Beasley
Numerade Educator
05:09

Problem 32

Compute the $\mathrm{L}^2$ inner product (3.16) and norms of the functions $f(x, y) \equiv 1$ and $g(x, y)=x^2+y^2$, when (a) $\Omega=\{0 \leq x \leq 1,0 \leq y \leq 1\}$ is the unit square;
(b) $\Omega=\left\{x^2+y^2 \leq 1\right\}$ is the unit disk.

Donald Yeh
Donald Yeh
Numerade Educator
01:46

Problem 32

Verify the triangle inequality for each pair of vectors in Exercise 3.2.1.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
07:55

Problem 32

Find all $\mathbf{v} \in \mathbb{R}^2$ such that
(a) $\|\mathbf{v}\|_1=\|\mathbf{v}\|_{\infty}$,
(b) $\|\mathbf{v}\|_1=\|\mathbf{v}\|_2$,
(c) $\|\mathbf{v}\|_2=\|\mathbf{v}\|_{\infty}$,
(d) $\|\mathbf{v}\|_{\infty}=\frac{1}{\sqrt{2}}\|\mathbf{v}\|_2$.

Gaurav Kalra
Gaurav Kalra
Numerade Educator

Problem 32

Let $K=A^T C A$, where $C>0$. Prove that
(a) $\operatorname{ker} K=\operatorname{coker} K=\operatorname{ker} A$;
(b) $\operatorname{img} K=\operatorname{coimg} K=\operatorname{coimg} A$.

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View

Problem 32

Prove that the space of complex $m \times n$ matrices is a complex vector space. What is its dimension?

Nick Johnson
Nick Johnson
Numerade Educator
12:06

Problem 33

Let $V$ be the vector space consisting of all continuous, vector-valued functions $\mathbf{f}(x)=\left(f_1(x), f_2(x)\right)^T$ defined on the interval $0 \leq x \leq 1$.
(a) Prove that $\langle\langle\mathbf{f}, \mathbf{g}\rangle\rangle=\int_0^1\left[f_1(x) g_1(x)+f_2(x) g_2(x)\right] d x$ defines an inner product on $V$.
(b) Prove, more generally, that if $\langle\mathbf{v}, \mathbf{w}\rangle$ is any inner product on $\mathbb{R}^2$, then
$\langle\langle\mathbf{f}, \mathbf{g}\rangle\rangle=\int_a^b\langle\mathbf{f}(x), \mathbf{g}(x)\rangle d x$ defines an inner product on $V$. (Part (a) corresponds to the dot product.) (c) Use part (b) to prove that
$$
\langle\langle\mathbf{f}, \mathbf{g}\rangle\rangle=\int_a^b\left[f_1(x) g_1(x)-f_1(x) g_2(x)-f_2(x) g_1(x)+3 f_2(x) g_2(x)\right] d x
$$
defines an inner product on $V$.

Chris Trentman
Chris Trentman
Numerade Educator
04:34

Problem 33

Verify the triangle inequality for the vectors and inner products in Exercise 3.2.4.

Ahmad Reda
Ahmad Reda
Numerade Educator
07:07

Problem 33

How would you quantify the following statement: The norm of a vector is small if and only if all its entries are small.

Shafiq Rehman
Shafiq Rehman
Numerade Educator
View

Problem 33

Prove that every positive definite matrix $K$ can be written as a Gram matrix.

Victor Salazar
Victor Salazar
Numerade Educator
02:33

Problem 33

Determine which of the following are subspaces of the vector space consisting of all complex $2 \times 2$ matrices. (a) All matrices with real diagonals. (b) All matrices for which the sum of the diagonal entries is zero. (c) All singular complex matrices. (d) All matrices whose determinant is real. (e) All matrices of the form $\left(\begin{array}{ll}a & \frac{b}{a} \\ b\end{array}\right)$, where $a, b \in \mathbb{C}$.

Daniel Pezzi
Daniel Pezzi
Numerade Educator
04:01

Problem 34

Verify the triangle inequality for the functions in Exercise 3.2 .12 for the indicated inner products.

Ahmad Reda
Ahmad Reda
Numerade Educator
05:54

Problem 34

Can you find an elementary proof of the inequalities $\|\mathbf{v}\|_{\infty} \leq\|\mathbf{v}\|_2 \leq \sqrt{n}\|\mathbf{v}\|_{\infty}$ for $\mathbf{v} \in \mathbb{R}^n$ directly from the formulas for the norms?

Victor Salazar
Victor Salazar
Numerade Educator
01:34

Problem 34

Suppose $K$ is the Gram matrix computed from $\mathbf{v}_1, \ldots, \mathbf{v}_n \in V$ relative to a given inner product. Let $\widetilde{K}$ be the Gram matrix for the same elements, but computed relative to a different inner product. Show that $K>0$ if and only if $\widetilde{K}>0$.

Narayan Hari
Narayan Hari
Numerade Educator
01:53

Problem 34

True or false: The set of all complex-valued functions $u(x)=v(x)+\mathrm{i} w(x)$ with $u(0)=\mathrm{i}$ is a subspace of the vector space of complex-valued functions.

Thomas Pauly
Thomas Pauly
Numerade Educator
02:29

Problem 35

Verify the triangle inequality for the two particular functions appearing in Exercise 3.1 .32 with respect to the $\mathrm{L}^2$ inner product on (a) the unit square; (b) the unit disk.

Vikash Ranjan
Vikash Ranjan
Numerade Educator
13:31

Problem 35

(i) Show the equivalence of the Euclidean norm and the 1 norm on $\mathbb{R}^n$ by proving $\|\mathbf{v}\|_2 \leq\|\mathbf{v}\|_1 \leq \sqrt{n}\|\mathbf{v}\|_2$. (ii) Verify that the vectors in Exercise 3.3 .31 satisfy both inequalities. (iii) For which vectors $\mathbf{v} \in \mathbb{R}^n$ is (a) $\|\mathbf{v}\|_2=\|\mathbf{v}\|_1$ ? (b) $\|\mathbf{v}\|_1=\sqrt{n}\|\mathbf{v}\|_2$ ?

Chris Trentman
Chris Trentman
Numerade Educator
04:37

Problem 35

Let $K_1=A_1^T C_1 A_1$ and $K_2=A_2^T C_2 A_2$ be any two $n \times n$ Gram matrices. Let $K=K_1+K_2$. (a) Show that if $K_1, K_2>0$ then $K>0$. (b) Give an example in which $K_1$ and $K_2$ are not positive definite, but $K>0$. (c) Show that $K$ is also a Gram matrix, by finding a matrix $A$ such that $K=A^T C A$.

Runpeng Li
Runpeng Li
Numerade Educator
01:05

Problem 35

Let $V$ denote the complex vector space spanned by the functions $1, e^{\mathrm{i} x}$ and $e^{-\mathrm{i} x}$, where $x$ is a real variable. Which of the following functions belong to $V$ ?
(a) $\sin x$,
(b) $\cos x-2 \mathrm{i} \sin x$,
(c) $\cosh x$,
(d) $\sin ^2 \frac{1}{2} x$,
(e) $\cos ^2 x ?$

Aman Gupta
Aman Gupta
Numerade Educator
01:31

Problem 36

Use the $\mathrm{L}^2$ inner product $\langle f, g\rangle=\int_{-1}^1 f(x) g(x) d x$ to answer the following:
(a) Find the "angle" between the functions 1 and $x$. Are they orthogonal? (b) Verify the Cauchy-Schwarz and triangle inequalities for these two functions. (c) Find all quadratic polynomials $p(x)=a+b x+c x^2$ that are orthogonal to both of these functions.

Amy Jiang
Amy Jiang
Numerade Educator
02:23

Problem 36

(i) Establish the equivalence inequalities (3.35) between the 1 and $\infty$ norms.
(ii) Verify them for the vectors in Exercise 3.3.31.
(iii) For which vectors $\mathbf{v} \in \mathbb{R}^n$ are your inequalities equality?

Ahmad Reda
Ahmad Reda
Numerade Educator
02:30

Problem 36

Show that $\mathbf{0} \neq \mathbf{z}$ is a null direction for the quadratic form $q(\mathbf{x})=\mathbf{x}^T K \mathbf{x}$ based on the Gram matrix $K=A^T C A$ if and only if $\mathbf{z} \in \operatorname{ker} K$.

Chris Trentman
Chris Trentman
Numerade Educator
01:02

Problem 36

Prove that the following define Hermitian inner products on $\mathbb{C}^2$ :
(a) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 \bar{w}_1+2 v_2 \bar{w}_2$,
(b) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 \bar{w}_1+\mathrm{i} v_1 \bar{w}_2-\mathrm{i} v_2 \bar{w}_1+2 v_2 \bar{w}_2$.

Victor Salazar
Victor Salazar
Numerade Educator
07:27

Problem 37

(a) Write down the explicit formulae for the Cauchy-Schwarz and triangle inequalities based on the weighted inner product $\langle f, g\rangle=\int_0^1 f(x) g(x) e^x d x$. (b) Verify that the inequalities hold when $f(x)=1, g(x)=e^x$ by direct computation. (c) What is the "angle" between these two functions in this inner product?

Ahmad Reda
Ahmad Reda
Numerade Educator

Problem 37

Let $\|\cdot\|_2$ denote the usual Euclidean norm on $\mathbb{R}^n$. Determine the constants in the norm equivalence inequalities $c^{\star}\|\mathbf{v}\| \leq\|\mathbf{v}\|_2 \leq C^{\star}\|\mathbf{v}\|$ for the following norms: (a) the weighted norm $\|\mathbf{v}\|=\sqrt{2 v_1^2+3 v_2^2}$,
(b) the norm $\|\mathbf{v}\|=\max \left\{\left|v_1+v_2\right|,\left|v_1-v_2\right|\right\}$.

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02:20

Problem 37

Which of the following define inner products on $\mathbb{C}^2$ ? (a) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 \bar{w}_1+2 \mathrm{i} v_2 \bar{w}_2$,
(b) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1+2 v_2 w_2$,
(c) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 \bar{w}_2+v_2 \bar{w}_1$,
(d) $\langle\mathbf{v}, \mathbf{w}\rangle=$
$2 v_1 \bar{w}_1+v_1 \bar{w}_2+v_2 \bar{w}_1+2 v_2 \bar{w}_2$,
(e) $\langle\mathbf{v}, \mathbf{w}\rangle=2 v_1 \bar{w}_1+(1+\mathrm{i}) v_1 \bar{w}_2+(1-\mathrm{i}) v_2 \bar{w}_1+3 v_2 \bar{w}_2$.

Wendi Zhao
Wendi Zhao
Numerade Educator
08:44

Problem 38

Answer Exercise 3.2.37 for the Sobolev $H^1$ inner product
$$
\langle f, g\rangle=\int_0^1\left[f(x) g(x)+f^{\prime}(x) g^{\prime}(x)\right] d x, \quad \text { cf. Exercise 3.1.27. }
$$

Ahmad Reda
Ahmad Reda
Numerade Educator

Problem 38

Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$. Prove that there is a constant $C>0$ such that the entries of every $\mathbf{v}=\left(v_1, v_2, \ldots, v_n\right)^T \in \mathbb{R}^n$ are all bounded, in absolute value, by $\left|v_i\right| \leq C\|\mathbf{v}\|$.

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Problem 38

Let $A=A^T$ be a real symmetric $n \times n$ matrix. Show that $(A \mathbf{v}) \cdot \mathbf{w}=\mathbf{v} \cdot(A \mathbf{w})$ for all $\mathbf{v}, \mathbf{w} \in \mathbb{C}^n$.

Nick Johnson
Nick Johnson
Numerade Educator
02:39

Problem 39

Prove that $\|\mathbf{v}-\mathbf{w}\| \geq|\|\mathbf{v}\|-\|\mathbf{w}\||$. Interpret this result pictorially.

Ziya Ogron
Ziya Ogron
Numerade Educator
04:10

Problem 39

Prove that if $[a, b]$ is a bounded interval and $f \in \mathrm{C}^0[a, b]$, then $\|f\|_2 \leq \sqrt{b-a}\|f\|_{\infty}$.

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator
02:14

Problem 39

Let $\mathbf{z}=\mathbf{x}+\mathrm{i} \mathbf{y} \in \mathbb{C}^n$.
(a) Prove that, for the Hermitian dot product, $\|\mathbf{z}\|^2=\|\mathbf{x}\|^2+\|\mathbf{y}\|^2$.
(b) Does this formula remain valid under a more general Hermitian inner product on $\mathbb{C}^n$ ?

Nick Johnson
Nick Johnson
Numerade Educator
00:34

Problem 40

True or false: $\|\mathbf{w}\| \leq\|\mathbf{v}\|+\|\mathbf{v}+\mathbf{w}\|$ for all $\mathbf{v}, \mathbf{w} \in V$.

Kayleah Tsai
Kayleah Tsai
Numerade Educator

Problem 40

In this exercise, the indicated function norms are taken over all of $\mathbb{R}$.
(a) Let $f_n(x)=\left\{\begin{array}{ll}1, & -n \leq x \leq n, \\ 0, & \text { otherwise. }\end{array}\right.$ Prove that $\left\|f_n\right\|_{\infty}=1$, but $\left\|f_n\right\|_2 \rightarrow \infty$ as $n \rightarrow \infty$.
(b) Explain why there is no constant $C$ such that $\|f\|_2 \leq C\|f\|_{\infty}$ for all functions $f$.
(c) Let $f_n(x)=\left\{\begin{array}{ll}\sqrt{\frac{n}{2}}, & -\frac{1}{n} \leq x \leq \frac{1}{n}, \\ 0, & \text { otherwise. }\end{array}\right.$ Prove that $\left\|f_n\right\|_2=1$, but $\left\|f_n\right\|_{\infty} \rightarrow \infty$ as $n \rightarrow \infty$. Conclude that there is no constant $C$ such that $\|f\|_{\infty} \leq C\|f\|_2$.
(d) Construct similar examples that disprove the related inequalities
(i) $\|f\|_{\infty} \leq C\|f\|_1$,
(ii) $\|f\|_1 \leq C\|f\|_2$,
(iii) $\|f\|_2 \leq C\|f\|_1$.

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15:25

Problem 40

Let $V$ be a complex inner product space. Prove that, for all $\mathbf{z}, \mathbf{w} \in V$,
(a) $\|\mathbf{z}+\mathbf{w}\|^2=\|\mathbf{z}\|^2+2 \operatorname{Re}\langle\mathbf{z}, \mathbf{w}\rangle+\|\mathbf{w}\|^2 ;$
(b) $\langle\mathbf{z}, \mathbf{w}\rangle=\frac{1}{4}\left(\|\mathbf{z}+\mathbf{w}\|^2-\|\mathbf{z}-\mathbf{w}\|^2+\mathrm{i}\|\mathbf{z}+\mathrm{i} \mathbf{w}\|^2-\mathrm{i}\|\mathbf{z}-\mathbf{i} \mathbf{w}\|^2\right)$.

Donald Albin
Donald Albin
Numerade Educator

Problem 41

(a) Prove that the space $\mathbb{R}^{\infty}$ consisting of all infinite sequences $\mathbf{x}=\left(x_1, x_2, x_3, \ldots\right)$ of real numbers $x_i \in \mathbb{R}$ is a vector space. (b) Prove that the set of all sequences $\mathbf{x}$ such that $\sum_{k=1}^{\infty} x_k^2<\infty$ is a subspace, commonly denoted by $\ell^2 \subset \mathbb{R}^{\infty}$. (c) Write down two examples of sequences $\mathrm{x}$ belonging to $\ell^2$ and two that do not belong to $\ell^2$. (d) True or false: If $\mathrm{x} \in \ell^2$, then $x_k \rightarrow 0$ and $k \rightarrow \infty$. (e) True or false: If $x_k \rightarrow 0$ as $k \rightarrow \infty$, then $\mathrm{x} \in \ell^2$. (f) Given $\alpha \in \mathbb{R}$, let $\mathbf{x}$ be the sequence with $x_k=\alpha^k$. For which values of $\alpha$ is $\mathbf{x} \in \ell^2$ ? (g) Answer part (f) when $x_k=k^\alpha$. (h) Prove that $\langle\mathbf{x}, \mathbf{y}\rangle=\sum_{k=1}^{\infty} x_k y_k$ defines an inner product on the vector space $\ell^2$. What is the corresponding norm? (i) Write out the Cauchy-Schwarz and triangle inequalities for the inner product space $\ell^2$.

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26:20

Problem 41

(a) Prove that the $\mathrm{L}^{\infty}$ and $\mathrm{L}^2$ norms on the vector space $\mathrm{C}^0[-1,1]$ are not equivalent.(b) Can you establish a bound in either direction, i.e., $\|f\|_{\infty} \leq C\|f\|_2$ or $\|f\|_2 \leq \bar{C}\|f\|_{\infty}$ for all $f \in \mathrm{C}^0[-1,1]$ for some positive constants $C, \bar{C}$ ? (c) Are the $\mathrm{L}^1$ and $\mathrm{L}^{\infty}$ norms equivalent?

Shafiq Rehman
Shafiq Rehman
Numerade Educator
16:09

Problem 41

(a) How would you define the angle between two elements of a complex inner product space? (b) What is the angle between $(-1,2-\mathrm{i},-1+2 \mathrm{i})^T$ and $(-2-\mathrm{i},-\mathrm{i}, 1-\mathrm{i})^T$ relative to the Hermitian dot product?

Donald Albin
Donald Albin
Numerade Educator
02:41

Problem 42

What does it mean if the constants defined in (3.36) are equal: $c^{\star}=C^{\star}$ ?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:14

Problem 42

Let $\mathbf{0} \neq \mathbf{v} \in \mathbb{C}^n$. Which scalar multiples $c \mathbf{v}$ have the same Hermitian norm as $\mathbf{v}$ ?

Nick Johnson
Nick Johnson
Numerade Educator
02:04

Problem 43

Suppose $\langle\mathbf{v}, \mathbf{w}\rangle_1$ and $\langle\mathbf{v}, \mathbf{w}\rangle_2$ are two inner products on the same vector space $V$. For which $\alpha, \beta \in \mathbb{R}$ is the linear combination $\langle\mathbf{v}, \mathbf{w}\rangle=\alpha\langle\mathbf{v}, \mathbf{w}\rangle_1+\beta\langle\mathbf{v}, \mathbf{w}\rangle_2$ a legitimate inner product?

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:51

Problem 43

Prove the Cauchy-Schwarz inequality (3.103) and the triangle inequality (3.104) for a general complex inner product.

Linda Winkler
Linda Winkler
Numerade Educator

Problem 44

Suppose $\|\cdot\|_1,\|\cdot\|_2$ are two norms on $\mathbb{R}^n$. Prove that the corresponding matrix norms satisfy $\widehat{c}^{\star}\|A\|_1 \leq\|A\|_2 \leq \hat{C}^{\star}\|A\|_1$ for any $n \times n$ matrix $A$ for some positive constants $0<\widehat{c}^{\star}<\hat{C}^{\star}$.

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01:02

Problem 44

The Hermitian adjoint of a complex $m \times n$ matrix $A$ is the complex conjugate of its transpose, written $A^{\dagger}=\overline{A^T}=\bar{A}^T$.
For example, if $A=\left(\begin{array}{cc}1+\mathrm{i} & 2 \mathrm{i} \\ -3 & 2-5 \mathrm{i}\end{array}\right)$, then $A^{\dagger}=\left(\begin{array}{cc}1-\mathrm{i} & -3 \\ -2 \mathrm{i} & 2+5 \mathrm{i}\end{array}\right)$. Prove that
(a) $\left(A^{\dagger}\right)^{\dagger}=A$,
(b) $(z A+w B)^{\dagger}=\bar{z} A^{\dagger}+\bar{w} B^{\dagger}$ for $z, w \in \mathbb{C}$,
(c) $(A B)^{\dagger}=B^{\dagger} A^{\dagger}$.

Victor Salazar
Victor Salazar
Numerade Educator
03:13

Problem 45

Compute the $\infty$ matrix norm of the following matrices.
(a) $\left(\begin{array}{ll}\frac{1}{2} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{6}\end{array}\right)$,
(b) $\left(\begin{array}{rr}\frac{5}{3} & \frac{4}{3} \\ -\frac{7}{6} & -\frac{5}{6}\end{array}\right)$,
(c) $\left(\begin{array}{rrr}0 & .1 & .8 \\ -.1 & 0 & .1 \\ -.8 & -.1 & 0\end{array}\right)$,
(d) $\left(\begin{array}{rrr}\frac{1}{3} & 0 & 0 \\ -\frac{1}{3} & 0 & \frac{1}{3} \\ 0 & \frac{2}{3} & \frac{1}{3}\end{array}\right)$.

Vishnu P
Vishnu P
Numerade Educator
01:00

Problem 45

A complex matrix $H$ is called Hermitian if it equals its Hermitian adjoint, $H^{\dagger}=H$, as defined in the preceding exercise. (a) Prove that the diagonal entries of a Hermitian matrix are real. (b) Prove that $(H \mathbf{z}) \cdot \mathbf{w}=\mathbf{z} \cdot(H \mathbf{w})$ for $\mathbf{z}, \mathbf{w} \in \mathbb{C}^n$. (c) Prove that every Hermitian inner product on $\mathbb{C}^n$ has the form $\langle\mathbf{z}, \mathbf{w}\rangle=\mathbf{z}^T H \overline{\mathbf{w}}$, where $H$ is an $n \times n$ positive definite Hermitian matrix. (d) How would you verify positive definiteness of a complex matrix?

Nick Johnson
Nick Johnson
Numerade Educator
02:32

Problem 46

Find a matrix $A$ such that $\left\|A^2\right\|_{\infty} \neq\|A\|_{\infty}^2$.

Manisha Sarker
Manisha Sarker
Numerade Educator
07:05

Problem 46

Multiple choice: Let $V$ be a complex normed vector space. How many unit vectors are parallel to a given vector $\mathbf{0} \neq \mathbf{v} \in V$ ? (a) none; (b) $1 ;$ (c) $2 ;$ (d) $3 ;(e) \infty$; (f) depends upon the vector; $(g)$ depends on the norm. Explain your answer.

Nasheed Jafri
Nasheed Jafri
Numerade Educator
03:33

Problem 47

True or false: If $B=S^{-1} A S$ are similar matrices, then $\|B\|_{\infty}=\|A\|_{\infty}$.

Patrick Burns
Patrick Burns
Numerade Educator
02:43

Problem 47

Let $\mathbf{v}_1, \ldots, \mathbf{v}_n$ be elements of a complex inner product space. Let $K$ denote the corresponding $n \times n$ Gram matrix, defined in the usual manner.
(a) Prove that $K$ is a Hermitian matrix, as defined in Exercise 3.6.45.
(b) Prove that $K$ is positive semi-definite, meaning $\mathbf{z}^T K \mathbf{z} \geq 0$ for all $\mathbf{z} \in \mathbb{C}^n$.
(c) Prove that $K$ is positive definite if and only if $\mathbf{v}_1, \ldots, \mathbf{v}_n$ are linearly independent.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 48

(i) Find an explicit formula for the 1 matrix norm $\|A\|_1$.
(ii) Compute the 1 matrix norm of the matrices in Exercise 3.3.45.

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08:49

Problem 48

For each of the following pairs of complex-valued functions,
(i) compute their $\mathrm{L}^2$ norm and Hermitian inner product on the interval $[0,1]$, and then (ii) check the validity of the Cauchy-Schwarz and triangle inequalities.
(a) $1, e^{i \pi x}$;
(b) $x+\mathrm{i}, x-\mathrm{i}$;
(c) $\mathrm{i} x^2,(1-2 \mathrm{i}) x+3 \mathrm{i}$.

Ahmad Reda
Ahmad Reda
Numerade Educator

Problem 49

Prove directly from the axioms of Definition 3.12 that (3.44) defines a norm on the space of $n \times n$ matrices.

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03:27

Problem 49

Formulate conditions on a weight function $w(x)$ that guarantee that the weighted integral $\langle f, g\rangle=\int_a^b f(x) \overline{g(x)} w(x) d x$ defines an inner product on the space of continuous complex-valued functions on $[a, b]$.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 50

Let $A=\left(\begin{array}{rr}1 & 1 \\ 1 & -2\end{array}\right)$. Compute the natural matrix norm $\|A\|$ for (a) the weighted $\infty$ norm $\|\mathbf{v}\|=\max \left\{2\left|v_1\right|, 3\left|v_2\right|\right\} ; \quad(b)$ the weighted 1 norm $\|\mathbf{v}\|=2\left|v_1\right|+3\left|v_2\right|$.

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13:31

Problem 50

(a) Formulate a general definition of a norm on a complex vector space.
(b) How would you define analogues of the $\mathrm{L}^1, \mathrm{~L}^2$ and $\mathrm{L}^{\infty}$ norms on $\mathbb{C}^n$ ?

Chris Trentman
Chris Trentman
Numerade Educator

Problem 51

The Frobenius norm of an $n \times n$ matrix $A$ is defined as $\|A\|_F=\sqrt{\sum_{i, j=1}^n a_{i j}^2}$. Prove that this defines a matrix norm by checking the three norm axioms plus the multiplicative inequality (3.42).

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Problem 52

Explain why $\|A\|=\max \left|a_{i j}\right|$ defines a norm on the space of $n \times n$ matrices. Show by example that this is not a matrix norm, i.e., (3.42) is not necessarily valid.

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