Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver, Chehrzad Shakiban

Chapter 7 Linearity - all with Video Answers

Educators

Chapter Questions

Problem 1

For each of the following linear transformations $L: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, find a matrix representative, and then describe its effect on (i) the $x$-axis; (ii) the unit square $S=\{0 \leq x, y \leq 1\}$; (iii) the unit disk $D=\left\{x^2+y^2 \leq 1\right\}$; (a) counterclockwise rotation by $45^{\circ}$; (b) rotation by $180^{\circ}$; (c) reflection in the line $y=2 x$; (d) shear along the $y$-axis of magnitude 2 ; (e) shear along the line $x=y$ of magnitude 3 ; (f) orthogonal projection on the line $y=2 x$.

Victor Salazar

Numerade Educator

02:24

Problem 1

True or false: An affine transformation takes (a) straight lines to straight lines;
(b) triangles to triangles; (c) squares to squares; (d) circles to circles; (e) ellipses to ellipses.

Suman Saurav Thakur

Numerade Educator

01:18

Problem 1

True or false: If $F[\mathbf{x}]$ is an affine transformation on $\mathbb{R}^n$, then the equation $F[\mathbf{x}]=\mathbf{c}$ defines a linear system.

Kayla Robinson

Numerade Educator

01:34

Problem 1

Which of the following functions $F: \mathbb{R}^3 \rightarrow \mathbb{R}$ are linear? (a) $F(x, y, z)=x$,
(b) $F(x, y, z)=y-2$, (c) $F(x, y, z)=x+y+3$, (d) $F(x, y, z)=x-y-z$,
(e) $F(x, y, z)=x y z$,
(f) $F(x, y, z)=x^2-y^2+z^2$,
(g) $F(x, y, z)=e^{x-y+z}$.

Wendi Zhao

Numerade Educator

00:33

Problem 1

Choose one from the following list of inner products on $\mathbb{R}^2$. Then find the adjoint of $A=\left(\begin{array}{rr}1 & 2 \\ -1 & 3\end{array}\right)$ when your inner product is used on both its domain and codomain. (a) the Euclidean dot product; (b) the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=2 v_1 w_1+3 v_2 w_2 ;$ (c) the inner product $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T K \mathbf{w}$ defined by the positive definite matrix $K=\left(\begin{array}{rr}2 & -1 \\ -1 & 4\end{array}\right)$.

Wendi Zhao

Numerade Educator

Problem 2

Let $L$ be the linear transformation represented by the matrix $\left(\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right)$. Show that $L^2=L \circ L$ is rotation by $180^{\circ}$. Is $L$ itself a rotation or a reflection?

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02:24

Problem 2

True or false: An affine transformation takes (a) straight lines to straight lines;
(b) triangles to triangles; $(c)$ squares to squares; (d) circles to circles; $(e)$ ellipses to ellipses.

Suman Saurav Thakur

Numerade Educator

04:12

Problem 2

(a) Let $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be an affine transformation. Let $L_1, L_2 \subset \mathbb{R}^n$ be two parallel lines. Prove that $F\left[L_1\right]$ and $F\left[L_2\right]$ are also parallel lines.
(b) Is the converse valid: if $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ maps parallel lines to parallel lines, then $F$ is necessarily an affine transformation?

Michael Jacobsen

Numerade Educator

Problem 2

Place each of the following linear systems in the form (7.45). Carefully describe the linear function, its domain, its codomain, and the right-hand side of the system. Which systems are homogeneous? (a) $3 x+5=0$, (b) $x=y+z$, (c) $a=2 b-3, b=c-1$, (d) $3(p-2)=2(q-3), p+q=0$, (e) $u^{\prime}+3 x u=0$, (f) $u^{\prime}+3 x=0$, (g) $u^{\prime}=u, u(0)=1$, (h) $u^{\prime \prime}-u=e^x, u(0)=3 u(1),(i) u^{\prime \prime}+x^2 u=3 x, u(0)=1, u^{\prime}(0)=0$, (j) $u^{\prime}=v$, $v^{\prime}=2 u,(k) u^{\prime \prime}-v^{\prime \prime}=2 u-v, u(0)=v(0), u(1)=v(1),(I) u(x)=1-3 \int_0^x u(y) d y$,
(m) $\int_0^{\infty} u(t) e^{-s t} d t=1+s^2, \quad(n) \int_0^1 u(x) d x=u\left(\frac{1}{2}\right), \quad(o) \int_0^1 u(y) d y=\int_0^1 y v(y) d y$, (p) $\frac{\partial u}{\partial t}+2 \frac{\partial u}{\partial x}=1$, (q) $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}, \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x},(r)-\frac{\partial^2 u}{\partial x^2}-\frac{\partial^2 u}{\partial y^2}=x^2+y^2-1$.

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02:41

Problem 2

Explain why the following functions $F: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ are not linear.
(a) $\left(\begin{array}{l}x+2 \\ x+y\end{array}\right)$,
(b) $\left(\begin{array}{l}x^2 \\ y^2\end{array}\right)$,
(c) $\left(\begin{array}{l}|y| \\ |x|\end{array}\right)$,
(d) $\left(\begin{array}{c}\sin (x+y) \\ x-y\end{array}\right)$,
(e) $\left(\begin{array}{l}x+e^y \\ 2 x+y\end{array}\right)$.

Gaurav Kalra

Numerade Educator

06:38

Problem 2

From the list in Exercise 7.5.1, choose different inner products on the domain and codomain, and then determine the adjoint of the matrix $A$.

Ahmad Reda

Numerade Educator

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Problem 3

Let $L$ be the linear transformation determined by $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$. Show $L^2=1$, and interpret geometrically.

Victor Salazar

Numerade Educator

01:41

Problem 3

Describe the image of (i) the $x$-axis, (ii) the unit disk $x^2+y^2 \leq 1$, (iii) the unit square $0 \leq x, y \leq 1$, under the following affine transformations:
(a) $T_1\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{r}2 \\ -1\end{array}\right)$,
(b) $T_2\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{ll}3 & 0 \\ 0 & 2\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{r}-1 \\ 0\end{array}\right)$,
(c) $T_3\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{l}1 \\ 2\end{array}\right)$,
(d) $T_4\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{l}1 \\ 0\end{array}\right)$,
(e) $T_5\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{rr}.6 & .8 \\ -.8 & .6\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{r}-3 \\ 2\end{array}\right)$,
(f) $T_6\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{cc}\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2}\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{l}1 \\ 0\end{array}\right)$.
(g) $T_7\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{rr}1 & 1 \\ -1 & 1\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{r}2 \\ -3\end{array}\right)$,
(h) $T_8\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{rr}2 & 1 \\ -2 & -1\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{l}1 \\ 1\end{array}\right)$.

Sriram Soundarrajan

Numerade Educator

00:37

Problem 3

The Fredholm Alternative of Theorem 4.46 first appeared in the study of what are now known as Fredholm integral equations: $u(x)+\int_a^b K(x, y) u(y) d y=f(x)$, in which $K(x, y)$ and $f(x)$ are prescribed continuous functions. Explain how the integral equation is a linear system; i.e, describe the linear map $L$, its domain and codomain, and prove linearity.

Frank Lin

Numerade Educator

02:41

Problem 3

Which of the following functions $F: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ are linear?
(a) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}x-y \\ x+y\end{array}\right)$,
(b) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}x+y+1 \\ x-y-1\end{array}\right)$,
(c) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}x y \\ x-y\end{array}\right)$,
(d) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}3 y \\ 2 x\end{array}\right)$,
(e) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}x^2+y^2 \\ x^2-y^2\end{array}\right)$,
(f) $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{c}y-3 x \\ x\end{array}\right)$.

Gaurav Kalra

Numerade Educator

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Problem 3

Choose one from the following list of inner products on $\mathbb{R}^3$ for both the domain and codomain, and find the adjoint of $A=\left(\begin{array}{rrr}1 & 1 & 0 \\ -1 & 0 & 1 \\ 0 & -1 & 2\end{array}\right)$ : (a) the Euclidean dot product;
Cumbuag Thasciagean (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 $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T K \mathbf{w}$ defined by the positive definite matrix $K=\left(\begin{array}{lll}2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2\end{array}\right)$.

Victor Salazar

Numerade Educator

03:58

Problem 4

What is the geometric interpretation of the linear transformation with matrix $A=\left(\begin{array}{rr}1 & 0 \\ 2 & -1\end{array}\right)$ ? Use this to explain why $A^2=1$.

Muhammad Saleem

Numerade Educator

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Problem 4

Using the affine transformations in Exercise 7.3.3, write out the following compositions and verify that they satisfy (7.37):
(a) $T_3 \circ T_4$,
(b) $T_4 \circ T_3$,
(c) $T_3 \circ T_6$,
(d) $T_6 \circ T_3$,
(e) $T_7 \circ T_8$,
(f) $T_8 \circ T_7$.

Nick Johnson

Numerade Educator

02:02

Problem 4

Answer Exercise 7.4.3 for the Volterra integral equation $u(t)+\int_a^t K(t, s) u(s) d s=f(t)$, where $a \leq t \leq b$.

Prachita Kush

Numerade Educator

01:02

Problem 4

Explain why the translation function $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, defined by $T\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}x+a \\ y+b\end{array}\right)$ for $\quad a, b \in \mathbb{R}$, is almost never linear. Precisely when is it linear? $a, b \in \mathbb{R}$, is almost never linear. Precisely when is it linear?

Monica Miller

Numerade Educator

02:36

Problem 4

From the list in Exercise 7.5.3, choose different inner products on the domain and codomain, and then compute the adjoint of the matrix $A$.

Jack Chen

Numerade Educator

01:27

Problem 5

Describe the image of the line $\ell$ that goes through the points $\left(\begin{array}{r}-2 \\ 1\end{array}\right),\left(\begin{array}{r}1 \\ -2\end{array}\right)$ under the linear transformation $\left(\begin{array}{rr}2 & 3 \\ -1 & 0\end{array}\right)$.

Vincenzo Zaccaro

Numerade Educator

02:06

Problem 5

Describe the image of the triangle with vertices $(-1,0),(1,0),(0,2)$ under the affine transformation $T\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{rr}4 & -1 \\ 2 & 5\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)+\left(\begin{array}{r}3 \\ -4\end{array}\right)$.

Erin Kirchmeier

Numerade Educator

01:09

Problem 5

(a) Prove that the solution to the linear integral equation $u(t)=a+\int_0^t k(s) u(s) d s$ solves the linear initial value problem $\frac{d u}{d t}=k(t) u(t), u(0)=a$.
(b) Use part (a) to solve the following integral equations
(i) $u(t)=2-\int_0^t u(s) d s$,
(ii) $u(t)=1+2 \int_1^t s u(s) d s$,
(iii) $u(t)=3+\int_0^t e^s u(s) d s$.

Fuzail Shakir

Numerade Educator

02:07

Problem 5

Find a matrix representation for the following linear transformations on $\mathbb{R}^3$ :
(a) counterclockwise rotation by $90^{\circ}$ around the $z$-axis; (b) clockwise rotation by $60^{\circ}$ around the $x$-axis; $(c)$ reflection through the $(x, y)$-plane; $(d)$ counterclockwise rotation by $120^{\circ}$ around the line $x=y=z ;(e)$ rotation by $180^{\circ}$ around the line $x=y=z$; $(f)$ orthogonal projection onto the $x y$-plane; $(g)$ orthogonal projection onto the plane $x-y+2 z=0$.

Victor Salazar

Numerade Educator

03:55

Problem 5

Choose an inner product on $\mathbb{R}^2$ from the list in Exercise 7.5.1, and an inner product on $\mathbb{R}^3$ from the list in Exercise 7.5.3, and then compute the adjoint of $A=\left(\begin{array}{rr}1 & 3 \\ 0 & 2 \\ -1 & 1\end{array}\right)$.

Ahmad Reda

Numerade Educator

02:26

Problem 6

Draw the parallelogram spanned by the vectors $\left(\begin{array}{l}1 \\ 2\end{array}\right)$ and $\left(\begin{array}{l}3 \\ 1\end{array}\right)$. Then draw its image under the linear transformations defined by the following matrices:
(a) $\left(\begin{array}{rr}1 & 0 \\ -1 & 1\end{array}\right)$,
(b) $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$,
(c) $\left(\begin{array}{rr}1 & 2 \\ -1 & 4\end{array}\right)$,
(d) $\left(\begin{array}{rr}\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\end{array}\right)$.
(c) $\left(\begin{array}{rr}-1 & -2 \\ 2 & 1\end{array}\right)$,
(f) $\left(\begin{array}{rr}\frac{1}{2} & -\frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2}\end{array}\right)$,
(g) $\left(\begin{array}{rr}2 & -1 \\ -4 & 2\end{array}\right)$.

Mahendra K

Numerade Educator

02:22

Problem 6

Under what conditions is the composition of two affine transformations
(a) a translation?
(b) a linear function?

Uma Kumari

Numerade Educator

00:59

Problem 6

Solve the following homogeneous linear ordinary differential equations. What is the dimension of the solution space? (a) $u^{\prime \prime}-4 u=0$, (b) $u^{\prime \prime}-6 u^{\prime}+8 u=0$,
(c) $u^{\prime \prime \prime}-9 u^{\prime}=0$,
(d) $u^{\prime \prime \prime \prime}+4 u^{\prime \prime \prime}-u^{\prime \prime}-16 u^{\prime}-12 u=0$.

Hunza Gilgit

Numerade Educator

07:52

Problem 6

Find a linear function $L: \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $L\left(\begin{array}{l}1 \\ 1\end{array}\right)=2$ and $L\left(\begin{array}{r}1 \\ -1\end{array}\right)=3$. Is it unique?

Anthony Ramos

Numerade Educator

02:38

Problem 6

Let $\mathcal{P}^{(2)}$ be the space of quadratic polynomials equipped with the inner product $\langle p, q\rangle=\int_0^1 p(x) q(x) d x$. Find the adjoint of the derivative operator $D[p]=p^{\prime}$ on $\mathcal{P}^{(2)}$.

Nick Johnson

Numerade Educator

01:54

Problem 7

Find a linear transformation that maps the unit circle $x^2+y^2=1$ to the ellipse $\frac{1}{4} x^2+\frac{1}{9} y^2=1$. Is your answer unique?

Lucas Finney

Numerade Educator

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Problem 7

(a) Under what conditions does an affine transformation have an inverse? (b) Is the inverse an affine transformation? If so, find a formula for its matrix and vector constituents. (c) Find the inverse, when it exists, of each of the the affine transformations in Exercise 7.3.3.

Victor Salazar

Numerade Educator

Problem 7

Define $L[y]=y^{\prime \prime}+y$. (a) Prove directly from the definition that $L: \mathrm{C}^2[a, b] \rightarrow \mathrm{C}^0[a, b]$ is a linear transformation. (b) Determine ker $L$.

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02:45

Problem 7

Find a linear function $L: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $L\left(\begin{array}{l}1 \\ 2\end{array}\right)=\left(\begin{array}{r}2 \\ -1\end{array}\right)$ and $L\left(\begin{array}{l}2 \\ 1\end{array}\right)=\left(\begin{array}{r}0 \\ -1\end{array}\right)$.

James Chok

Numerade Educator

01:01

Problem 7

Prove that, if it exists, the adjoint of a linear function is uniquely determined by (7.80).

Raj Bala

Numerade Educator

00:57

Problem 8

Find a linear transformation that maps the unit sphere $x^2+y^2+z^2=1$ to the ellipsoid $x^2+\frac{1}{4} y^2+\frac{1}{16} z^2=1$

Linh Vu

Numerade Educator

05:31

Problem 8

Let $\mathrm{v}_1, \ldots, \mathrm{v}_n$ be a basis for $\mathbb{R}^n$. (a) Show that every affine transformation $F[\mathbf{x}]=A \mathbf{x}+\mathbf{b}$ on $\mathbb{R}^n$ is uniquely determined by the $n+1$ vectors $\mathbf{w}_0=F[\mathbf{0}], \mathbf{w}_1=$ $F\left[\mathbf{v}_1\right], \ldots$,
$\mathbf{w}_n=F\left[\mathbf{v}_n\right]$. (b) Find the formula for $A$ and $\mathbf{b}$ when $\mathbf{v}_1=\mathbf{e}_1, \ldots, \mathbf{v}_n=\mathbf{e}_n$ are the standard basis vectors. (c) Find the formula for $A, \mathbf{b}$ for a general basis $\mathbf{v}_1, \ldots, \mathbf{v}_n$.

Henry Carnick

Numerade Educator

03:48

Problem 8

Answer Exercise 7.4.7 when $L=3 D^2-2 D-5$.

Ryan Mcalister

Numerade Educator

00:49

Problem 8

Under what conditions does there exist a linear function $L: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $L\left(\begin{array}{l}x_1 \\ y_1\end{array}\right)=\left(\begin{array}{l}a_1 \\ b_1\end{array}\right)$ and $L\left(\begin{array}{l}x_2 \\ y_2\end{array}\right)=\left(\begin{array}{l}a_2 \\ b_2\end{array}\right)$ ? Under what conditions is $L$ uniquely defined? In the latter case, write down the matrix representation of $L$.

Ankit Gupta

Numerade Educator

03:26

Problem 8

Prove that
(a) $(L+M)^*=L^*+M^*$,
(b) $(c L)^*=c L^*$ for $c \in \mathbb{R}$,
(c) $\left(L^*\right)^*=L$,
(d) $\left(L^{-1}\right)^*=\left(L^*\right)^{-1}$.

Peiling Liu

Numerade Educator

02:00

Problem 9

True or false: A linear transformation $L: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ maps
(a) straight lines to straight lines; (b) triangles to triangles; (c) squares to squares;
(d) circles to circles; (e) ellipses to ellipses.

Patrick Vaughn

Numerade Educator

03:05

Problem 9

Show that the space of all affine transformations on $\mathbb{R}^n$ is a vector space. What is its dimension?

Nick Johnson

Numerade Educator

00:53

Problem 9

Consider the linear differential equation $y^{\prime \prime \prime}+5 y^{\prime \prime}+3 y^{\prime}-9 y=0$. (a) Write the equation in the form $L[y]=0$ for a differential operator $L=p(D)$. (b) Find a basis for ker $L$, and then write out the general solution to the differential equation.

Mike Gaerlan

Numerade Educator

02:23

Problem 9

Can you construct a linear function $L: \mathbb{R}^3 \rightarrow \mathbb{R}$ such that $L\left(\begin{array}{r}1 \\ -1 \\ 0\end{array}\right)=1, L\left(\begin{array}{r}1 \\ 0 \\ -1\end{array}\right)=4$, and $L\left(\begin{array}{r}0 \\ 1 \\ -1\end{array}\right)=-2$ ? If yes, find one. If not, explain why not.

Kayla Robinson

Numerade Educator

07:24

Problem 9

Let $L: U \rightarrow V$ be a linear function between inner product spaces. Prove that $\mathbf{u} \in \mathbb{R}^n$ solves the inhomogeneous linear system $L[\mathbf{u}]=\mathbf{f}$ if and only if
$$
\left\langle\mathbf{u}, L^*[\mathbf{v}]\right\rangle=\langle\mathbf{f}, \mathbf{v}\rangle \quad \text { for all } \quad \mathbf{v} \in V .
$$
Explain why Exercise 3.1.11 is a special case of this result. Remark. Equation (7.83) is known as the weak formulation of the linear system. It plays an essential role in the analysis of differential equations and their numerical approximations, [61].

Ed Tam

Numerade Educator

02:24

Problem 10

(a) Prove that the linear transformation associated with the improper orthogonal matrix $\left(\begin{array}{rr}\cos \theta & \sin \theta \\ \sin \theta & -\cos \theta\end{array}\right)$ is a reflection through the line that makes an angle $\frac{1}{2} \theta$ with the $x$-axis.
(b) Show that the composition of two such reflections, with angles $\theta, \varphi$, is a rotation.
What is the angle of the rotation? Does the composition depend upon the order of the two reflections?

Victor Salazar

Numerade Educator

02:10

Problem 10

In this exercise, we establish a useful matrix representation for affine transformations. We identify $\mathbb{R}^n$ with the $n$-dimensional affine subspace (as in Exercise 2.2.28)
$$
V_n=\left\{(\mathbf{x}, 1)^T=\left(x_1, \ldots, x_n, 1\right)^T\right\} \subset \mathbb{R}^{n+1}
$$
consisting of vectors whose last coordinate is fixed at $x_{n+1}=1$. (a) Show that multiplication of vectors $\left(\begin{array}{c}\mathbf{x} \\ 1\end{array}\right) \in V_n$ by the $(n+1) \times(n+1)$ affine matrix $\left(\begin{array}{cc}A & \mathbf{b} \\ 0 & 1\end{array}\right)$ coincides with the action (7.35) of an affine transformation on $\mathbf{x} \in \mathbb{R}^n$. (b) Prove that the composition law (7.37) for affine transformations corresponds to multiplication of their affine matrices. (c) Define the inverse of an affine transformation in the evident manner, and show that it corresponds to the inverse affine matrix.

Ashley Boni

Numerade Educator

01:05

Problem 10

The following functions are solutions to a constant coefficient homogeneous scalar ordinary differential equation. (i) Determine the least possible order of the differential equation, and (ii) write down an appropriate differential equation.
(a) $e^{2 x}+e^{-3 x}$,
(b) $1+e^{-x}$,
(c) $x e^x$,
(d) $e^x+2 e^{2 x}+3 e^{3 x}$.

Srikar Pasumarthy

Numerade Educator

04:43

Problem 10

Given $\mathbf{a}=(a, b, c)^T \in \mathbb{R}^3$, prove that the cross product map $L_{\mathbf{a}}[\mathbf{v}]=\mathbf{a} \times \mathbf{v}$, as defined in (4.2), is linear, and find its matrix representative.

Jose Hannan

Numerade Educator

Problem 10

Suppose $V, W$ are finite-dimensional inner product spaces with dual space $V^*, W^*$. Let $L: V \rightarrow W$ be a linear function, and let $\bar{L}^*: W^* \rightarrow V^*$ denote the dual linear function, as in Exercise 7.2 .30 (without the tilde), while $L^*: W \rightarrow V$ denotes its adjoint. (As noted above, the same notation denotes two mathematically different objects.) Prove that if we identify $V^* \simeq V$ and $W^* \simeq W$ using the linear isomorphism in Exercise 7.1.62, then the dual and adjoint functions are identified $\bar{L}^* \simeq L^*$, thus reconciling the unfortunate clash in notation. In particular, this includes the two possible interpretations of the transpose of a matrix.

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01:05

Problem 11

(a) Find the matrix in $\mathbb{R}^3$ that corresponds to a counterclockwise rotation around the $x$-axis through an angle $60^{\circ}$. (b) Write it as a. product of elementary matrices, and interpret each of the factors.

Raj Bala

Numerade Educator

07:11

Problem 11

Which of the indicated maps $\mathbf{F}(x, y)$ define isometries of the Euclidean plane?
(a) $\left(\begin{array}{c}y \\ -x\end{array}\right)$,
(b) $\left(\begin{array}{l}x-2 \\ y-1\end{array}\right)$
(c) $\left(\begin{array}{c}x-y+1 \\ x+2\end{array}\right)$.
(d) $\frac{1}{\sqrt{2}}\left(\begin{array}{l}x+y-3 \\ x+y-2\end{array}\right)$,
(e) $\frac{1}{5}\left(\begin{array}{c}3 x+4 y \\ -4 x+3 y+1\end{array}\right)$.

Ahmad Reda

Numerade Educator

13:12

Problem 11

Solve the following Euler differential equations:
(a) $x^2 u^{\prime \prime}+5 x u^{\prime}-5 u=0$,
(b) $2 x^2 u^{\prime \prime}-x u^{\prime}-2 u=0$,
(c) $x^2 u^{\prime \prime}-u=0$,
(d) $x^2 u^{\prime \prime}+x u^{\prime}-3 u=0$,
(e) $3 x^2 u^{\prime \prime}-5 x u^{\prime}-3 u=0$,
(f) $\frac{d^2 u}{d x^2}+\frac{2}{x} \frac{d u}{d x}=0$.

Khoobchandra Agrawal

Numerade Educator

Problem 11

Is the Euclidean norm function $N(\mathbf{v})=\|\mathbf{v}\|$, for $\mathbf{v} \in \mathbb{R}^n$, linear?

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Problem 11

Show that the following linear transformations of $\mathbb{R}^2$ are self-adjoint with respect to the Euclidean dot product: (a) rotation through the angle $\theta=\pi$; (b) reflection about the line $y=x$. (c) The scaling map $S[\mathbf{x}]=3 \mathbf{x} ;(d)$ orthogonal projection onto the line $y=x$.

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Problem 12

Let $L \subset \mathbb{R}^2$ be the line through the origin in the direction of a unit vector $\mathbf{u}$. (a) Prove that the matrix representative of reflection through $L$ is $R=2 \mathbf{u u}^T-1$. (b) Find the corresponding formula for reflection through the line in the direction of a general nonzero vector $\mathbf{v} \neq \mathbf{0}$. (c) Determine the matrix representative for reflection through the line in the direction $(i)(1,0)^T$,
(ii) $\left(\frac{3}{5},-\frac{4}{5}\right)^T$,
(iii) $(1,1)^T$,
(iv) $(2,-3)^T$.

Victor Salazar

Numerade Educator

Problem 12

Prove that the planar affine isometry $F\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{r}-y+1 \\ x-2\end{array}\right)$ represents a rotation through an angle of $90^{\circ}$ around the center $\left(\frac{3}{2},-\frac{1}{2}\right)^T$.

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02:37

Problem 12

Solve the third order Euler differential equation $x^3 u^{\prime \prime \prime}+2 x^2 u^{\prime \prime}-3 x u^{\prime}+3 u=0$ by using the power ansatz (7.56). What is the dimension of the solution space for $x>0$ ? For all $x$ ?

Jacob Denson

Numerade Educator

07:24

Problem 12

Let $V$ be a vector space. Prove that every linear function $L: \mathbb{R} \rightarrow V$ has the form $L[x]=x \mathbf{b}$, where $x \in \mathbb{R}$, for some $\mathbf{b} \in V$.

Ed Tam

Numerade Educator

13:34

Problem 12

Let $M$ be a positive definite matrix. Show that $A: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is self-adjoint with respect to the inner product $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T M \mathbf{w}$ if and only if $M A$ is a symmetric matrix.

Anthony Ramos

Numerade Educator

Problem 13

Decompose the following matrices into a product of elementary matrices. Then interpret each of the factors as a linear transformation.
(a) $\left(\begin{array}{rr}0 & 2 \\ -3 & 1\end{array}\right)$,
(b) $\left(\begin{array}{rr}1 & 1 \\ -1 & 1\end{array}\right)$,
(c) $\left(\begin{array}{ll}3 & 1 \\ 1 & 2\end{array}\right)$,
(d) $\left(\begin{array}{lll}1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1\end{array}\right)$,
(e) $\left(\begin{array}{lll}1 & 2 & 0 \\ 2 & 4 & 1 \\ 2 & 1 & 1\end{array}\right)$.

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00:48

Problem 13

True or false: The map $L[\mathbf{x}]=-\mathbf{x}$ for $\mathbf{x} \in \mathbb{R}^n$ defines (a) an isometry; (b) a rotation.

Subhadeepta Sahoo

Numerade Educator

03:48

Problem 13

(i) Show that if $u(x)$ solves the Euler equation $a x^2 \frac{d^2 u}{d x^2}+b x \frac{d u}{d x}+c u=0$, then $v(t)=u\left(e^t\right)$ solves a linear, constant coefficient differential equation. (ii) Use this alternative technique to solve the Euler differential equations in Exercise

Ashley Boni

Numerade Educator

Problem 13

True or false: The quadratic form $Q(\mathbf{v})=\mathbf{v}^T K \mathbf{v}$ defined by a symmetric $n \times n$ matrix $K$ defines a linear function $Q: \mathbb{R}^n \rightarrow \mathbb{R}$.

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00:31

Problem 13

Prove that $A=\left(\begin{array}{ll}6 & 3 \\ 2 & 4\end{array}\right)$ is self-adjoint with respect to the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=2 v_1 w_1+3 v_2 w_2$. Hint: Use the criterion in Exercise 7.5.12.

Wendi Zhao

Numerade Educator

03:52

Problem 14

(a) Prove that $\left(\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)=\left(\begin{array}{ll}1 & a \\ 0 & 1\end{array}\right)\left(\begin{array}{ll}1 & 0 \\ b & 1\end{array}\right)\left(\begin{array}{ll}1 & a \\ 0 & 1\end{array}\right)$, where $a=-\tan \frac{1}{2} \theta$ and $b=\sin \theta . \quad(b)$ Is the factorization valid for all values of $\theta$ ? (c) Interpret the factorization geometrically. Remark. The factored version is less prone to numerical errors due to round-off, and so can be used when extremely accurate numerical computations involving rotations are required.

Urvashi Arora

Numerade Educator

Problem 14

Prove that every proper affine plane isometry $F[\mathbf{x}]=Q \mathbf{x}+\mathbf{b}$ of $\mathbb{R}^2$, where $\operatorname{det} Q=1$, is either (i) a translation, or (ii) a rotation (7.43) centered at some point $\mathbf{c} \in \mathbb{R}^2$.
Hint: Use Exercise 1.5.7.

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Problem 14

(a) Use the method in Exercise 7.4.13 to solve an Euler equation whose characteristic equation has a double root $r_1=r_2=r$. (b) Solve the specific equations
(i) $x^2 u^{\prime \prime}-x u^{\prime}+u=0$,
(ii) $\frac{d^2 u}{d x^2}+\frac{1}{x} \frac{d u}{d x}=0$.

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04:09

Problem 14

(a) Prove that $L$ is linear if and only if it satisfies (7.3).
(b) Use induction to prove that $L$ satisfies (7.4).

Himanshu Kushwaha

Numerade Educator

00:26

Problem 14

Consider the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1+\frac{1}{2} v_2 w_2+\frac{1}{3} v_3 w_3$ on $\mathbb{R}^3$.
(a) What are the conditions on the entries of a $3 \times 3$ matrix $A$ in order that it be selfadjoint? Hint: Use the criterion in Exercise 7.5.12. (b) Write down an example of a non-diagonal self-adjoint matrix.

Wendi Zhao

Numerade Educator

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Problem 15

Determine the matrix representative for orthogonal projection $P: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ on the line through the origin in the direction $(a)(1,0)^T,(b)(1,1)^T,(c)(2,-3)^T$.

Victor Salazar

Numerade Educator

01:06

Problem 15

Compute both compositions $F \circ G$ and $G \circ F$ of the following affine transformations on $\mathrm{R}^2$. Which pairs commute? (a) $F=$ counterclockwise rotation around the origin by $45^{\circ}$; $G=$ translation in the $y$ direction by 3 units. (b) $F=$ counterclockwise rotation around the point $(1,1)^T$ by $30^{\circ} ; G=$ counterclockwise rotation around the point $(-2,1)^T$ by $90^{\circ}$. (c) $F=$ reflection through the line $y=x+1 ; G=$ rotation around $(1,1)^T$ by $180^{\circ}$.

Brandon Fox

Numerade Educator

03:15

Problem 15

Show that if $u(x)$ solves $x u^{\prime \prime}+2 u^{\prime}-4 x u=0$, then $v(x)=x u(x)$ solves a linear, constant coefficient equation. Use this to find the general solution to the given differential equation. Which of your solutions are continuous at the singular point $x=0$ ? Differentiable?

Charles Machakwa

Numerade Educator

02:43

Problem 15

Answer Exercise 7.5.14 for the inner product based on $\left(\begin{array}{rrr}2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right)$.

Patrick Burns

Numerade Educator

07:23

Problem 15

Let $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right), B=\left(\begin{array}{ll}p & q \\ r & s\end{array}\right)$ be $2 \times 2$ matrices. For each of the following functions, prove that $L: \mathcal{M}_{2 \times 2} \rightarrow \mathcal{M}_{2 \times 2}$ defines a linear map, and then find its matrix representative with respect to the standard basis $\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right),\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right),\left(\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right),\left(\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right)$ of $\mathcal{M}_{2 \times 2}$ :
(a) $L[X]=A X$,
(b) $R[X]=X B$,
(c) $K[X]=A X B$.

Anthony Ramos

Numerade Educator

Problem 16

(a) Prove that every $2 \times 2$ matrix of rank 1 can be written in the form $A=\mathbf{u} \mathbf{v}^T$ where $\mathbf{u}, \mathbf{v} \in \mathbb{R}^2$ are non-zero column vectors. (b) Which rank one matrices correspond to orthogonal projection onto a one-dimensional subspace of $\mathbb{R}^2$ ?

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Problem 16

In $\mathrm{R}^2$, show the following: (a) The composition of two affine isometries is another affine isometry. (b) The composition of two translations is another translation. (c) The composition of a translation and a rotation (not necessarily centered at the origin) in either order is a rotation. (d) The composition of two plane rotations is either another rotation or a translation. What is the condition for the latter possibility? (e) Every plane translation can be written as the composition of two rotations.

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05:25

Problem 16

Let $S \subset \mathbb{R}$ be an open subset (i.e., a union of open intervals), and let $D: \mathrm{C}^1(S) \rightarrow$ $\mathrm{C}^0(S)$ be the derivative operator $D[f]=f^{\prime}$. True or false: $\operatorname{ker} D$ is a one-dimensional subspace of $\mathrm{C}^1(S)$.

Mengchun Cai

Numerade Educator

00:49

Problem 16

True or false: The identity transformation is self-adjoint for an arbitrary inner product on the underlying vector space.

Dharmendra Jain

Numerade Educator

00:33

Problem 16

The domain space of the following functions is the space of $n \times n$ real matrices $A$. Which are linear? What is the codomain space in each case? (a) $L[A]=3 A$;
(b) $L[A]=\mathrm{I}-A ;$ (c) $L[A]=A^T ;$ (d) $L[A]=A^{-1} ;$ (e) $L[A]=\operatorname{det} A ;$ (f) $L[A]=\operatorname{tr} A$;
(g) $L[A]=\left(a_{11}, \ldots, a_{n n}\right)^T$, i.e., the vector of diagonal entries of $A$;
(h) $L[A]=A \mathbf{v}$, where $\mathbf{v} \in \mathbb{R}^n ;$ (i) $L[A]=\mathbf{v}^T A \mathbf{v}$, where $\mathbf{v} \in \mathbb{R}^n$.

Monica Miller

Numerade Educator

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Problem 17

Give a geometrical interpretation of the linear transformations on $\mathbb{R}^3$ defined by each of the six $3 \times 3$ permutation matrices (1.30).

Victor Salazar

Numerade Educator

02:47

Problem 17

Let $\ell$ be a line in $\mathbb{R}^2$. A glide reflection is an affine map on $\mathbb{R}^2$ composed of a translation in the direction of $\ell$ by a distance $d$ followed by a reflection through $\ell$. Find the formula for a glide reflection along
(a) the $x$-axis by a distance 2 ; (b) the line $y=x$ by a distance 3 in the direction of increasing $x$; $(c)$ the line $x+y=1$ by a distance 2 in the direction of increasing $x$.

Jay Patel

Numerade Educator

02:14

Problem 17

Show that $\log \left(x^2+y^2\right)$ and $\frac{x}{x^2+y^2}$ are harmonic functions, that is, solutions of the two-dimensional Laplace equation.

Amit Srivastava

Numerade Educator

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Problem 17

True or false: A diagonal matrix is self-adjoint for an arbitrary inner product on $\mathbb{R}^n$.

Donna Densmore

Numerade Educator

04:47

Problem 17

Let $\mathbf{v}_1, \ldots, \mathbf{v}_n$ be a basis of $V$ and $\mathbf{w}_1, \ldots, \mathbf{w}_n$ be any vectors in $W$. Show that there is a unique linear function $L: V \rightarrow W$ such that $L\left[\mathbf{v}_i\right]=\mathbf{w}_i, i=1, \ldots, n$.

Ernest Castorena

Numerade Educator

07:36

Problem 18

Write down the $3 \times 3$ matrix $X_\psi$ representing a clockwise rotation in $\mathbb{R}^3$ around the $x$-axis by angle $\psi$.

Anthony Ramos

Numerade Educator

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Problem 18

Let $\ell$ be the line in the direction of the unit vector $\mathbf{u}$ through the point $\mathbf{a}$. (a) Write down the formula for the affine map defining the reflection through the line $\ell$. Hint: Use Exercise 7.2.12. (b) Write down the formula for the glide reflection, as defined in Exercise 7.3.17, along $\ell$ by a distance $d$ in the direction of $\mathbf{u}$. (c) Prove that every improper affine plane isometry is either a reflection or a glide reflection. Hint: Use Exercise 7.2.10.

Victor Salazar

Numerade Educator

02:17

Problem 18

Find all solutions $u=f(r)$ of the two-dimensional Laplace equation that depend only on the radial coordinate $r=\sqrt{x^2+y^2}$. Do these solutions form a vector space? If so, what is its dimension?

Manik Pulyani

Numerade Educator

01:57

Problem 18

Suppose $L: U \rightarrow U$ has an adjoint $L^*: U \rightarrow U$. (a) Show that $L+L^*$ is self-adjoint.
(b) Show that $L \circ L^*$ is self-adjoint.

Nick Johnson

Numerade Educator

03:04

Problem 18

Bilinear functions: Let $V, W, Z$ be vector spaces. A function that takes any pair of vectors $\mathbf{v} \in V$ and $\mathbf{w} \in W$ to a vector $\mathbf{z}=B[\mathbf{v}, \mathbf{w}] \in Z$ is called bilinear if, for each fixed $\mathbf{w}$, it is a linear function of $\mathbf{v}$, so $B[c \mathbf{v}+d \overline{\mathbf{v}}, \mathbf{w}]=c B[\mathbf{v}, \mathbf{w}]+d B[\overline{\mathbf{v}}, \mathbf{w}]$, and, for each fixed $\mathbf{v}$, it is a linear function of $\mathbf{w}$, so $B[\mathbf{v}, c \mathbf{w}+d \overline{\mathbf{w}}]=c B[\mathbf{v}, \mathbf{w}]+d B[\mathbf{v}, \overline{\mathbf{w}}]$.
Thus, $B: V \times W \rightarrow Z$ defines a function on the Cartesian product space $V \times W$, as defined in Exercise 2.1.13. (a) Show that $B[\mathbf{v}, \mathbf{w}]=v_1 w_1-2 v_2 w_2$ is a bilinear function from $\mathbb{R}^2 \times \mathbb{R}^2$ to $\mathbb{R}$. (b) Show that $B[\mathbf{v}, \mathbf{w}]=2 v_1 w_2-3 v_2 w_3$ is a bilinear function from $\mathbb{R}^2 \times \mathbb{R}^3$ to $\mathbb{R}$. (c) Show that if $V$ is an inner product space, then $B[\mathbf{v}, \mathbf{w}]=\langle\mathbf{v}, \mathbf{w}\rangle$ defines a bilinear function $B: V \times V \rightarrow \mathbb{R}$. (d) Show that if $A$ is any $m \times n$ matrix, then $B[\mathbf{v}, \mathbf{w}]=\mathbf{v}^T A \mathbf{w}$ defines a bilinear function $B: \mathbb{R}^m \times \mathbb{R}^n \rightarrow \mathbb{R}$. (e) Show that every bilinear function $B: \mathbb{R}^m \times \mathbb{R}^n \rightarrow \mathbb{R}$ arises in this way. (f) Show that a vector-valued function $B: \mathbb{R}^m \times \mathbb{R}^n \rightarrow \mathbb{R}^k$ defines a bilinear function if and only if each of its components $B_i: \mathbb{R}^m \times \mathbb{R}^n \rightarrow \mathbb{R}$, for $i=1, \ldots, k$, is a bilinear function. $(g)$ True or false: A bilinear function $B: V \times W \rightarrow Z$ defines a linear function on the Cartesian product space.

Nick Johnson

Numerade Educator

00:15

Problem 19

Explain why the linear map defined by - I defines a rotation in two-dimensional space, but a reflection in three-dimensional space.

Amrita Bhasin

Numerade Educator

02:24

Problem 19

A set of $n+1$ points $\mathbf{a}_0, \ldots, \mathbf{a}_n \in \mathbb{R}^n$ is said to be in general position if the differences $\mathbf{a}_i-\mathbf{a}_j \operatorname{span} \mathbb{R}^n$. (a) Show that the points are in general position if and only if they do not all lie in a proper affine subspace $A \subseteq \mathbb{R}^n$, of. Exercise 2.2 .28 . (b) Let $\mathrm{a}_0, \ldots, \mathrm{a}_n$ and $\mathbf{b}_0, \ldots, \mathbf{b}_n$ be two sets in general position. Show that there is an isometry $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $F\left[\mathbf{a}_i\right]=\mathbf{b}_i$ for all $i=0, \ldots, n$, if and only if their interpoint distances agree: $\mid \mathbf{a}_i-\mathbf{a}_j\|=\| \mathbf{b}_i-\mathbf{b}_j \|$ for all $0 \leq i<j \leq n$. Hint: Use Exercise 4.3.19.

Daniel Pezzi

Numerade Educator

01:02

Problem 19

Find all (real) solutions to the two-dimensional Laplace equation of the form $u=\log p(x, y)$, where $p(x, y)$ is a quadratic polynomial. Do these solutions form a vector space? If so, what is its dimension?

Carson Merrill

Numerade Educator

Problem 19

Suppose $J, M: U \rightarrow U$ are self-adjoint linear functions on an inner product space $U$.
(a) Prove that $\langle J[\mathbf{u}], \mathbf{u}\rangle=\langle M[\mathbf{u}], \mathbf{u}\rangle$ for all $\mathbf{u} \in U$ if and only if $J=M$.
(b) Explain why this result is false if the self-adjointness hypothesis is dropped.

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01:06

Problem 19

Which of the following define linear operators on the vector space $C^1(\mathbb{R})$ of continuously differentiable scalar functions? What is the codomain?
(a) $L[f]=f(0)+f(1)$,
(b) $L[f]=f(0) f(1)$,
(c) $L[f]=f^{\prime}(1)$
(d) $L[f]=f^{\prime}(3)-f(2)$,
(e) $L[f]=x^2 f(x)$,
(f) $L[f]=f(x+2)$
(g) $L[f]=f(x)+2$,
(h) $L[f]=f^{\prime}(2 x)$,
(i) $L[f]=f^{\prime}\left(x^2\right)$,
(j) $L[f]=f(x) \sin x-f^{\prime}(x) \cos x$,
(k) $L[f]=2 \log f(0)$,
(l) $L[f]=\int_0^1 e^y f(y) d y$,
(m) $L[f]=\int_0^1|f(y)| d y$,
(n) $L[f]=\int_{x-1}^{x+1} f(y) d y$,
(o) $L[f]=\int_x^{x^2} \frac{f(y)}{y} d y$,
(p) $L[f]=\int_0^{f(x)} y d y$,
(q) $L[f]=\int_0^x y^2 f^{\prime}(y) d y$,
(r) $L[f]=\int_{-1}^1[f(y)-f(0)] d y$,
(s) $L[f]=\int_{-1}^x[f(y)-y] d y$.

Aman Gupta

Numerade Educator

00:38

Problem 20

Let $\mathbf{u}=\left(u_1, u_2, u_3\right)^T \in \mathbb{R}^3$ be a unit vector. Show that $Q_\pi=2 \mathbf{u} \mathbf{u}^T-\mathbf{I}$ represents rotation around the axis $\mathbf{u}$ through an angle $\pi$.

Jeyasree R T

Numerade Educator

Problem 20

Suppose that $V$ is an inner product space and $L: V \rightarrow V$ is an isometry, so $\|L[\mathbf{v}]\|=\|\mathbf{v}\|$ for all $\mathbf{v} \in V$. Prove that $L$ also preserves the inner product: $\langle L[\mathbf{v}], L[\mathbf{w}]\rangle=\langle\mathbf{v}, \mathbf{w}\rangle$. Hint: Look at $\|L[\mathbf{v}+\mathbf{w}]\|^2$.

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04:19

Problem 20

(a) Show that the function $e^x \cos y$ is a solution to the two-dimensional Laplace equation. (b) Show that its quadratic Taylor polynomial at $x=y=0$ is harmonic.
(c) What about its degree 3 Taylor polynomial? (d) Can you state a general theorem?
(e) Test your result by looking at the Taylor polynomials of the harmonic function
$$
\log \left[(x-1)^2+y^2\right]
$$

James Kiss

Numerade Educator

01:52

Problem 20

Prove that if $L: U \rightarrow U$ is an invertible linear transformation on an inner product space $U$, then the following three statements are equivalent: (a) $\langle L[\mathbf{u}], L[\mathbf{v}]\rangle=\langle\mathbf{u}, \mathbf{v}\rangle$ for all $\mathbf{u}, \mathbf{v} \in U$. (b) $\|L[\mathbf{u}]\|=\|\mathbf{u}\|$ for all $\mathbf{u} \in U$. (c) $L^*=L^{-1}$. Hint: Use Exercise 7.5.19.

Nick Johnson

Numerade Educator

01:10

Problem 20

True or false: The average or mean $A[f]=\frac{1}{b-a} \int_a^b f(x) d x$ of a function on the interval $[a, b]$ defines a linear operator $A: \mathrm{C}^0[a, b] \rightarrow \mathbb{R}$.

Linda Hand

Numerade Educator

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Problem 21

Let $\mathbf{u} \in \mathbb{R}^3$ be a unit vector. (a) Explain why the elementary reflection matrix $R=\mathrm{I}-2 \mathbf{u} \mathbf{u}^T$ represents a reflection through the plane orthogonal to $\mathbf{u}$. (b) Prove that $R$ is an orthogonal matrix. Is it proper or improper? (c) Write out $R$ when $\mathbf{u}=$ (i) $\left(\frac{3}{5}, 0,-\frac{4}{5}\right)^T$, (ii) $\left(\frac{3}{13}, \frac{4}{13},-\frac{12}{13}\right)^T$, (iii) $\left(\frac{1}{\sqrt{6}},-\frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}\right)^T$.
(d) Give a geometrical explanation why $Q_\pi=-R$ represents the rotation of Exercise 7.2 .20 .

Victor Salazar

Numerade Educator

Problem 21

Let $V$ be a normed vector space. Prove that a linear map $L: V \rightarrow V$ defines an isometry of $V$ for the given norm if and only if it maps the unit sphere $S_1=\{\|\mathbf{u}\|=1\}$ to itself: $L\left[S_1\right]=\left\{L[\mathbf{u}] \mid \mathbf{u} \in S_1\right\}=S_1$.

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02:06

Problem 21

(a) Find a basis for, and the dimension of, the vector space consisting of all quadratic polynomial solutions of the three-dimensional Laplace equation $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}=0$.
(b) Do the same for the homogeneous cubic polynomial solutions.

Bcrypt_Sha256$$2B$12$Jyg5Xsmd/D90Hrerlbrjb.Vvn1Eec8Hpzl08Ivucuckdn8Igliwh6 Bcrypt_Sha256$$2B$12$Jyg5Xsmd/D90Hrerlbrjb.Yfcjxtp7V4Zxsmgv8Xpg.Vn.Fy6Khx6

Numerade Educator

Problem 21

(a) Prove that the operation $M_a[u(x)]=a(x) u(x)$ of multiplication by a continuous function $a(x)$ defines a self-adjoint linear operator on the function space $\mathrm{C}^0[a, b]$ with respect to the $\mathrm{L}^2$ inner product. (b) Is $M_a$ also self-adjoint with respect to the weighted inner product $\langle\langle f, g\rangle\rangle=\int_a^b f(x) g(x) w(x) d x$ ?

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01:44

Problem 21

Prove that multiplication $M_h[f(x)]=h(x) f(x)$ by a given function $h \in \mathrm{C}^n[a, b]$ defines a linear operator $M_h: \mathrm{C}^n[a, b] \rightarrow \mathrm{C}^n[a, b]$. Which result from calculus do you need to complete the proof?

Mohamed Mohamed

Numerade Educator

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Problem 22

Let $\mathbf{a} \in \mathbb{R}^3$, and let $Q$ be any $3 \times 3$ rotation matrix such that $Q \mathbf{a}=\mathbf{e}_3$. (a) Show, using the notation of (7.25), that $R_\theta=Q^T Z_\theta Q$ represents rotation around a by angle $\theta$.
(b) Verify this formula in the case $\mathbf{a}=\mathbf{e}_2$ by comparing with (7.26).

Victor Salazar

Numerade Educator

Problem 22

(a) List all linear and affine isometries of $\mathbb{R}^2$ with respect to the $\infty$ norm. Hint: Use Exercise 7.3.21. (b) Can you generalize your results to $\mathbb{R}^3$ ?

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06:06

Problem 22

Find all solutions $u=f(r)$ of the three-dimensional Laplace equation $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+\frac{\partial^2 u}{\partial z^2}=0$ that depend only on the radial coordinate $r=\sqrt{x^2+y^2+z^2}$.
Do these solutions form a vector space? If so, what is its dimension?

Keran Li

Numerade Educator

Problem 22

A linear function $S: U \rightarrow U$ is called skew-adjoint if $S^*=-S$. (a) Prove that a skew-symmetric matrix is skew-adjoint with respect to the standard dot product on $\mathbb{R}^n$. (b) Under what conditions is $S[\mathbf{x}]=A \mathbf{x}$ skew-adjoint with respect to the inner product $\langle\mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^T M \mathbf{y}$ on $\mathbb{R}^n$ ? (c) Let $L: U \rightarrow U$ have an adjoint $L^*$. Prove that $L-L^*$ is skew-adjoint. (d) Explain why every linear operator $L: U \rightarrow U$ that has an adjoint $L^*$ can be written as the sum of a self-adjoint and a skew-adjoint operator.

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01:29

Problem 22

Show that if $w(x)$ is any continuous function, then the weighted integral
$$
I_w[f]=\int_a^b f(x) w(x) d x \text { defines a linear operator } I_w: \mathrm{C}^0[a, b] \rightarrow \mathbb{R} .
$$

Adrian Co

Numerade Educator

03:31

Problem 23

Quaternions: The skew field $\mathrm{H}$ of quaternions can be identified with the vector space $\mathbb{R}^4$ equipped with a noncommutative multiplication operation. The standard basis vectors $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3, \mathbf{e}_4$ are traditionally denoted by the letters $1, \mathrm{i}, \mathrm{j}, \mathrm{k} ;$ the vector $(a, b, c, d)^T \in \mathbb{R}^4$ corresponds to the quaternion $q=a+b i+c j+d k$. Quaternion addition coincides with vector addition. Quaternion multiplication is defined so that
$$
1 q=q=q 1, \mathrm{i}^2=\mathrm{j}^2=\mathrm{k}^2=-1, \mathrm{i} \mathrm{j}=\mathrm{k}=-\mathrm{j} \mathrm{i}, \mathrm{i} \mathrm{k}=-\mathrm{j}=-\mathrm{ki}, \mathrm{jk}=\mathrm{i}=-\mathrm{k} \mathrm{j},
$$
along with the distributive laws
$$
(q+r) s=q s+r s, \quad q(r+s)=q r+q s, \quad \text { for all } \quad q, r, s \in \mathbb{H} \text {. }
$$

Nick Johnson

Numerade Educator

Problem 23

Answer Exercise 7.3.22 for the 1 norm.

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Problem 23

Let $L, M$ be linear functions. (a) Prove that $\operatorname{ker}(L \circ M) \supseteq \operatorname{ker} M$. (b) Find an example in which $\operatorname{ker}(L \circ M) \neq \operatorname{ker} M$.

Victor Salazar

Numerade Educator

Problem 23

(a) Let $L_1: U \rightarrow V_1$ and $L_2: U \rightarrow V_2$ be linear maps between inner product spaces, with $V_1, V_2$ not necessarily the same. Let $J_1=L_1^* \circ L_1, J_2=L_2^* \circ L_2$. Show that the sum $J=J_1+J_2$ can be written as a self-adjoint combination $J=L^* \circ L$ for some linear operator

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01:54

Problem 23

(a) Show that the partial derivatives $\partial_x[f]=\frac{\partial f}{\partial x}$ and $\partial_y[f]=\frac{\partial f}{\partial y}$ both define linear operators on the space of continuously differentiable functions $f(x, y)$.
(b) For which values of $a, b, c, d$ is the map $L[f]=a \frac{\partial f}{\partial x}+b \frac{\partial f}{\partial y}+c f+d$ linear?

Vikash Ranjan

Numerade Educator

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Problem 24

Find the matrix form of the linear transformation $L(x, y)=\left(\begin{array}{c}x-4 y \\ -2 x+3 y\end{array}\right)$ with respect to the following bases of $\mathbb{R}^2$ :
(a) $\left(\begin{array}{l}1 \\ 0\end{array}\right),\left(\begin{array}{l}0 \\ 1\end{array}\right)$,
(b) $\left(\begin{array}{l}2 \\ 0\end{array}\right),\left(\begin{array}{l}0 \\ 3\end{array}\right)$,
(c) $\left(\begin{array}{l}1 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ 1\end{array}\right)$,
(d) $\left(\begin{array}{l}2 \\ 1\end{array}\right)+\left(\begin{array}{r}-1 \\ 1\end{array}\right)$,
(e) $\left(\begin{array}{l}3 \\ 2\end{array}\right),\left(\begin{array}{l}2 \\ 3\end{array}\right)$.

Victor Salazar

Numerade Educator

03:17

Problem 24

A matrix of the form $H=\left(\begin{array}{ll}\cosh \alpha & \sinh \alpha \\ \sinh \alpha & \cosh \alpha\end{array}\right)$ for $\alpha \in \mathbb{R}$ defines a hyperbolic rotation of $\mathbb{R}^2$. (a) Prove that all hyperbolic rotations preserve the indefinite quadratic form $q(\mathbf{x})=x^2-y^2$ in the sense that $q(H \mathbf{x})=q(\mathbf{x})$ for all $\mathbf{x}=(x, y)^T \in \mathbb{R}^2$. Observe that ordinary rotations preserve circles $x^2+y^2=a$, while hyperbolic rotations preserve: hyperbolas $x^2-y^2=a$. (b) Are there any other affine transformations of $\mathbb{R}^2$ that preserve the quadratic form $q(\mathbf{x})$ ? Remark. The four-dimensional version of this construction, i.e., affine maps preserving the indefinite Minkowski form $t^2-x^2-y^2-z^2$, forms the geometrical foundation for Einstein's theory of special relativity, [55].

Chai Santi

Numerade Educator

05:08

Problem 24

For each of the following inhomogeneous systems, determine whether the right-hand side lies in the image of the coefficient matrix, and, if so, write out the general solution, clearly identifying the particular solution and the kernel element.
(a) $\left(\begin{array}{ll}1 & -1 \\ 3 & -3\end{array}\right) \mathrm{x}=\left(\begin{array}{l}1 \\ 2\end{array}\right)$,
(b) $\left(\begin{array}{rrr}2 & 1 & 4 \\ -1 & 2 & 1\end{array}\right) \mathbf{x}=\left(\begin{array}{l}1 \\ 2\end{array}\right)$,
(c) $\left(\begin{array}{rrr}1 & 2 & -1 \\ 2 & 0 & 1 \\ 1 & -2 & 2\end{array}\right) \mathbf{x}=\left(\begin{array}{l}0 \\ 3 \\ 3\end{array}\right)$.
(d) $\left(\begin{array}{rr}-2 & 1 \\ -2 & 3 \\ 3 & -5\end{array}\right) \mathbf{x}=\left(\begin{array}{r}1 \\ 0 \\ -1\end{array}\right)$,
(e) $\left(\begin{array}{rrrr}-1 & 3 & 0 & 2 \\ 2 & -6 & 1 & -1 \\ -3 & 9 & -2 & 0\end{array}\right) \mathbf{x}=\left(\begin{array}{r}2 \\ -2 \\ 2\end{array}\right)$.

Jingyun Wang

Numerade Educator

05:47

Problem 24

Find the minimum value of $p(\mathbf{u})=\frac{1}{2} \mathbf{u}^T\left(\begin{array}{rr}3 & -2 \\ -2 & 3\end{array}\right) \mathbf{u}-\mathbf{u}^T\left(\begin{array}{r}1 \\ -1\end{array}\right)$ for $\mathbf{u} \in \mathbb{R}^2$.

Patrick Hall

Numerade Educator

01:01

Problem 24

Prove that the Laplacian operator $\Delta[f]=\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}$ defines a linear function on the vector space of twice continuously differentiable functions $f(x, y)$.

Hast Aggarwal

Numerade Educator

01:04

Problem 25

Let $\ell \subset \mathbb{R}^2$ be a line, and $\mathbf{p} \notin \ell$ a point. A perspective map takes a point $x \in \mathbb{R}^2$ to the point $q \in \ell$ that is the intersection of $\ell$ with the line going through $\mathbf{p}$ and $\mathbf{x}$. If the line is parallel to $\ell$, then the map is not defined. Find the formula for the perspective map when (a) $\ell$ is the $x$-axis and $\mathrm{p}=(0,1)^T,(b) \ell$ is the line $y=x$ and $\mathbf{p}=(1,0)^T$. Is either map affine? An isometry? Remark. Mapping three-dimensional objects onto a two-dimensional screen (or your retina) is based on perspective maps, which are thus of fundamental importance in art, optics, computer vision, computer graphics and animation, and computer games.

Carson Merrill

Numerade Educator

Problem 25

Which of the following systems have a unique solution?
(a) $\left(\begin{array}{rr}3 & 1 \\ -1 & -1 \\ 2 & 0\end{array}\right)\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}0 \\ 2 \\ 2\end{array}\right)$,
(b) $\left(\begin{array}{rrr}1 & 2 & -1 \\ -2 & 3 & 0\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{l}1 \\ 2\end{array}\right)$.
(c) $\left(\begin{array}{rrr}2 & 1 & -1 \\ 0 & -3 & -3 \\ 2 & 0 & -2\end{array}\right)\left(\begin{array}{c}u \\ v \\ w\end{array}\right)=\left(\begin{array}{r}3 \\ -1 \\ 5\end{array}\right)$,
(d) $\left(\begin{array}{lll}1 & 4 & -1 \\ 1 & 3 & -3 \\ 2 & 3 & -2\end{array}\right)\left(\begin{array}{l}u \\ v \\ w\end{array}\right)=\left(\begin{array}{r}-2 \\ -1 \\ 1\end{array}\right)$.

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02:40

Problem 25

Minimize the function $p(\mathbf{u})=\frac{1}{2} \mathbf{u}^T\left(\begin{array}{rrr}2 & -1 & 0 \\ -1 & 4 & -2 \\ 0 & -2 & 3\end{array}\right) \mathbf{u}-\mathbf{u}^T\left(\begin{array}{r}2 \\ 0 \\ -1\end{array}\right)$ for $\mathbf{u} \in \mathbb{R}^3$.

Noah Musser

Numerade Educator

02:55

Problem 25

Show that the gradient $G[f]=\nabla f$ defines a linear operator from the space of continuously differentiable scalar-valued functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ to the space of continuous vector fields $\mathbf{v}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$.

Harshita Goel

Numerade Educator

01:05

Problem 26

. Find the matrix form of $L[\mathbf{x}]=\left(\begin{array}{lll}-3 & 2 & 2 \\ -3 & 1 & 3 \\ -1 & 2 & 0\end{array}\right) \times$ with respect to the following bases of $\mathbb{R}^3$ :
(a) $\left(\begin{array}{l}2 \\ 0 \\ 0\end{array}\right),\left(\begin{array}{r}0 \\ -1 \\ 0\end{array}\right),\left(\begin{array}{r}0 \\ 0 \\ -2\end{array}\right)$,
(b)
$\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right),\left(\begin{array}{r}0 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right)$
(c) $\left(\begin{array}{l}2 \\ 1 \\ 2\end{array}\right),\left(\begin{array}{r}0 \\ 1 \\ -1\end{array}\right),\left(\begin{array}{r}1 \\ -2 \\ 1\end{array}\right)$.

James Kiss

Numerade Educator

01:34

Problem 26

Find bases of the domain and codomain that place the following matrices in the canonical form (7.33). Use (7.32) to check your answer.
(a) $\left(\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right)$,
(b) $\left(\begin{array}{rrr}1 & -3 & 4 \\ -2 & 6 & -8\end{array}\right),(c)\left(\begin{array}{rr}2 & 3 \\ 0 & 4 \\ -1 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}1 & 2 & 1 \\ 1 & -1 & -1 \\ 2 & 1 & 0\end{array}\right)$,
(e) $\left(\begin{array}{rrrr}1 & 3 & 0 & 1 \\ 2 & 6 & 1 & -2 \\ -1 & -3 & -1 & 3 \\ 0 & 0 & -1 & 4\end{array}\right)$.

Anthony Ramos

Numerade Educator

02:00

Problem 26

Solve the following inhomogeneous linear ordinary differential equations:
(a) $u^{\prime}-4 u=x-3$,
(b) $5 u^{\prime \prime}-4 u^{\prime}+4 u=e^x \cos x$,
(c) $u^{\prime \prime}-3 u^{\prime}=e^{3 x}$.

Anand Jangid

Numerade Educator

03:08

Problem 26

Minimize $\mid(2 x-y, x+y)^T \|^2-6 x$ over all $x, y$, where $\|\cdot\|$ denotes the Euclidean norm on $\mathbb{R}^2$.

Noah Musser

Numerade Educator

04:55

Problem 26

Prove that, on $\mathbb{R}^3$, the gradient, curl, and divergence all define linear operators. Be precise in your description of the domain space and the codomain space in each case.

Harshita Goel

Numerade Educator

Problem 27

(a) Show that every invertible linear function $L: \mathbb{R}^n \rightarrow \mathbb{R}^n$ can be represented by the identity matrix by choosing appropriate (and not necessarily the same) bases on the domain and codomain. (b) Which linear transformations are represented by the identity matrix when the domain and codomain are required to have the same basis? (c) Find bases of $\mathbb{R}^2$ so that the following linear transformations are represented by the identity matrix: (i) the scaling map $S[\mathbf{x}]=2 \mathrm{x} ;$ (ii) counterclockwise rotation by $45^{\circ} ;$ (iii) the shear $\left(\begin{array}{rr}1 & 0 \\ -2 & 1\end{array}\right)$.

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01:14

Problem 27

Answer Exercise 7.5.26 for (a) the weighted norm $\left\|(x, y)^T\right\|=\sqrt{2 x^2+3 y^2}$;
(b) the norm based on $\left(\begin{array}{rr}2 & -1 \\ -1 & 1\end{array}\right)$; (c) the norm based on $\left(\begin{array}{ll}3 & 1 \\ 1 & 3\end{array}\right)$.

Manik Pulyani

Numerade Educator

Problem 27

Write down a basis for and dimension of the linear function spaces (a) $\mathcal{L}\left(\mathbb{R}^3, \mathbb{R}\right)$,
(b) $\mathcal{L}\left(\mathbb{R}^2, \mathbb{R}^2\right)$, (c) $\mathcal{L}\left(\mathbb{R}^m, \mathbb{R}^n\right)$,
(d) $\mathcal{L}\left(\mathcal{P}^{(3)}, \mathbb{R}\right)$,
(e) $\mathcal{L}\left(\mathcal{P}^{(2)}, \mathbb{R}^2\right)$,
(f) $\mathcal{L}\left(\mathcal{P}^{(2)}, \mathcal{P}^{(2)}\right)$.
Here $\mathcal{P}^{(n)}$ is the space of polynomials of degree $\leq n$.

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02:43

Problem 28

Suppose a linear transformation $L: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is represented by a symmetric matrix with respect to the standard basis $\mathbf{e}_1, \ldots, \mathbf{e}_n$. (a) Prove that its matrix representative with respect to any orthonormal basis $\mathbf{u}_1, \ldots, \mathbf{u}_n$ is symmetric. (b) Is it symmetric when expressed in terms of a non-orthonormal basis?

Nick Johnson

Numerade Educator

01:49

Problem 28

Solve the following initial value problems: (a) $u^{\prime}+3 u=e^x, u(1)=0$, (b) $u^{\prime \prime}+4 u=1$,
$$
\begin{aligned}
& u(\pi)=u^{\prime}(\pi)=0, \quad(c) u^{\prime \prime}-u^{\prime}-2 u=e^x+e^{-x}, u(0)=u^{\prime}(0)=0, \quad(d) u^{\prime \prime}+2 u^{\prime}+5 u=\sin x, \\
& u(0)=1, u^{\prime}(0)=0, \quad(e) u^{\prime \prime \prime}-u^{\prime \prime}+u^{\prime}-u=x, u(0)=0, u^{\prime}(0)=1, u^{\prime \prime}(0)=0 .
\end{aligned}
$$

Linh Vu

Numerade Educator

14:29

Problem 28

Solve the following inhomogeneous Euler equations using either variation of parameters or the change of variables method discussed in Exercise 7.4.13:
(a) $x^2 u^{\prime \prime}+x u^{\prime}-u=x$,
(b) $x^2 u^{\prime \prime}-2 x u^{\prime}+2 u=\log x$,
(c) $x^2 u^{\prime \prime}-3 x u^{\prime}-5 u=3 x-5$.

Nadir Musofer

Numerade Educator

02:07

Problem 28

Let $L(x, y)=\left(\begin{array}{c}x-2 y \\ x+y \\ -x+3 y\end{array}\right)$ and $\mathbf{f}=\left(\begin{array}{l}1 \\ 0\end{array}\right)$. Minimize $p(\mathbf{x})=\frac{1}{2}\|L[\mathbf{x}]\|^2-\langle\mathbf{x}, \mathbf{f}\rangle$ using (a) the Euclidean inner products and norms on both $\mathbb{R}^2$ and $\mathbb{R}^3$; (b) the Euclidean inner product on $\mathbb{R}^2$ and the weighted norm $\|\mathbf{w}\|=\sqrt{w_1^2+2 w_2^2+3 w_3^2}$ on $\mathbb{R}^3 ;$ (c) the inner product given by $\left(\begin{array}{rr}2 & -1 \\ -1 & 2\end{array}\right)$ on $\mathbb{R}^2$ and the Euclidean norm on $\mathbb{R}^3 ;(d)$ the inner product. given by $\left(\begin{array}{rr}2 & -1 \\ -1 & 2\end{array}\right)$ on $\mathbb{R}^2$ and the weighted norm $\|\mathbf{w}\|=\sqrt{w_1^2+2 w_2^2+3 w_3^2}$ on $\mathbb{R}^3$.

Noah Musser

Numerade Educator

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Problem 28

True or false: The set of linear transformations $L: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $L\left(\begin{array}{l}1 \\ 0\end{array}\right)=\left(\begin{array}{l}0 \\ 0\end{array}\right)$ is a subspace of $\mathcal{L}\left(\mathbb{R}^2, \mathbb{R}^2\right)$. If true, what is its dimension? is a subspace of $\mathcal{L}\left(\mathbb{R}^3, \mathbb{R}^3\right)$. If true, what is its dimension?

Victor Salazar

Numerade Educator

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Problem 29

In this exercise, we show that every inner product $\langle\cdot, \cdot\rangle$ on $\mathbb{R}^n$ can be reduced to the dot product when expressed in a suitably adapted basis. (a) Specifically, prove that there exists a basis $\mathbf{v}_1, \ldots, \mathbf{v}_n$ of $\mathbb{R}^n$ such that $\langle\mathbf{x}, \mathbf{y}\rangle=\sum_{i=1}^n c_i d_i=\mathbf{c} \cdot \mathbf{d}$, where $\mathbf{c}=\left(c_1, c_2, \ldots, c_n\right)^T$ are the coordinates of $\mathbf{x}$ and $\mathbf{d}=\left(d_1, d_2, \ldots, d_n\right)^T$ those of $\mathbf{y}$ with respect to the basis. Is the basis uniquely determined? (b) Find bases that reduce the following inner products to the dot product on $\mathbb{R}^2$ :
(i) $\langle\mathbf{v}, \mathbf{w}\rangle=2 v_1 w_1+3 v_2 w_2$,
(ii) $\langle\mathbf{v}, \mathbf{w}\rangle=v_1 w_1-v_1 w_2-v_2 w_1+3 v_2 w_2$.

Victor Salazar

Numerade Educator

Problem 29

Write down all solutions to the following boundary value problems. Label your answer as (i) unique solution, (ii) no solution, (iii) infinitely many solutions.
(a) $u^{\prime \prime}+2 u=2 x, u(0)=0, u(\pi)=0$,
(b) $u^{\prime \prime}+4 u=\cos x, u(-\pi)=0, u(\pi)=1$,
(c) $u^{\prime \prime}-2 u^{\prime}+u=x-2, u(0)=-1, u(1)=1$,
(d) $u^{\prime \prime}+2 u^{\prime}+2 u=1, u(0)=\frac{1}{2}, u(\pi)=\frac{1}{2}$,
(e) $u^{\prime \prime}-3 u^{\prime}+2 u=4 x, u(0)=0, u(1)=0$,
(f) $x^2 u^{\prime \prime}+x u^{\prime}-u=0, u(0)=1, u(1)=0$,
(g) $x^2 u^{\prime \prime}-6 u=0, u(1)=1, u(2)=-1$,
(h) $x^2 u^{\prime \prime}-2 x u^{\prime}+2 u=0, u(0)=0, u(1)=1$.

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03:08

Problem 29

Find the minimum distance between the point $(1,0,0)^T$ and the plane $x+y-z=0$ when distance is measured in (a) the Euclidean norm; (b) the weighted norm $\|\mathbf{w}\|=$ $\sqrt{w_1^2+2 w_2^2+3 w_3^2} ;$ (c) the norm based on the positive definite matrix $\left(\begin{array}{rrr}3 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 3\end{array}\right)$.

Lucas Finney

Numerade Educator

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Problem 29

True or false: The set of linear transformations $L: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ such that $L\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)=\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)$ is a subspace of $\mathcal{L}\left(\mathbb{R}^3, \mathbb{R}^3\right)$. If true, what is its dimension?

Victor Salazar

Numerade Educator

Problem 30

Dual functions: Let $L: V \rightarrow W$ be a linear function between vector spaces. The dual linear function, denoted by $L^*: W^* \rightarrow V^*$ (note the change in direction) is defined so that $L^*(m)=m \circ L$ for all linear functions $m \in W^*$. (a) Prove that $L^*$ is a linear function. (b) If $M: W \rightarrow Z$ is linear, prove that $(M \circ L)^*=L^* \circ M^*$. (c) Suppose $\operatorname{dim} V=n$ and $\operatorname{dim} W=m$. Prove that if $L$ is represented by the $m \times n$ matrix $A$ with respect to bases of $V, W$, then $L^*$ is represented by the $n \times m$ transposed matrix $A^T$ with respect to the dual bases, as defined in Exercise 7.1.32.

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22:21

Problem 30

Let $L: U \rightarrow V$ be a linear function, and let $W \subset U$ be a subspace of the domain space. (a) Prove that $Y=\{L[\mathbf{w}] \mid \mathbf{w} \in W\} \subset \operatorname{img} L \subset V$ is a subspace of the image.
(b) Prove that $\operatorname{dim} Y \leq \operatorname{dim} W$. Conclude that a linear transformation can never increase the dimension of a subspace.

Anthony Ramos

Numerade Educator

04:59

Problem 30

How would you modify the statement of Theorem 7.62 if $\operatorname{ker} L \neq\{0\}$ ?

Patrick Vaughan

Numerade Educator

00:33

Problem 30

Consider the linear function $L: \mathbb{R}^3 \rightarrow \mathbb{R}$ defined by $L(x, y, z)=3 x-y+2 z$. Write down the vector $\mathbf{a} \in \mathbb{R}^3$ such that $L[\mathbf{v}]=\langle\mathbf{a}, \mathbf{v}\rangle$ when the inner product is (a) the Euclidean 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 defined by the positive definite matrix $K=\left(\begin{array}{rrr}2 & -1 & 0 \\ -1 & 2 & 1 \\ 0 & 1 & 2\end{array}\right)$.

Wendi Zhao

Numerade Educator

02:43

Problem 31

Suppose $A$ is an $m \times n$ matrix. (a) Let $\mathbf{v}_1, \ldots, \mathbf{v}_n$ be a basis of $\mathbb{R}^n$, and $A \mathbf{v}_i=\mathbf{w}_i \in \mathbb{R}^m$, for $i=1, \ldots, n$. Prove that the vectors $\mathbf{v}_1, \ldots, \mathbf{v}_n, \mathbf{w}_1, \ldots, \mathbf{w}_n$, serve to uniquely specify $A$.
(b) Write down a formula for $A$.

Ed Tam

Numerade Educator

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Problem 31

(a) Show that if $L: V \rightarrow V$ is linear and $\operatorname{ker} L \neq\{0\}$, then $L$ is not invertible.
(b) Show that if img $L \neq V$, then $L$ is not invertible.
(c) Give an example of a linear map with $\operatorname{ker} L=\{0\}$ that is not invertible. Hint: First explain why your example must be on an infinite-dimensional vector space.

Victor Salazar

Numerade Educator

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Problem 31

Let $\mathbb{R}^n$ be equipped with the inner product $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^T K \mathbf{w}$. Let $L[\mathbf{v}]=\mathbf{r} \mathbf{v}$ where $\mathbf{r}$ is a row vector of size $1 \times n$. (a) Find a formula for the column vector a such that (7.12) holds for the linear function $L: \mathbb{R}^n \rightarrow \mathbb{R}$. (b) Illustrate your result when $\mathbf{r}=(2,-1)$, using (i) the dot product (ii) the weighted inner product $\langle\mathbf{v}, \mathbf{w}\rangle=3 v_1 w_1+2 v_2 w_2$, (iii) the inner product induced by $K=\left(\begin{array}{rr}2 & -1 \\ -1 & 3\end{array}\right)$.

Victor Salazar

Numerade Educator

13:02

Problem 32

Use superposition to solve the following inhomogeneous ordinary differential equations:
(a) $u^{\prime}+2 u=1+\cos x$,
(b) $u^{\prime \prime}-9 u=x+\sin x$,
(c) $9 u^{\prime \prime}-18 u^{\prime}+10 u=1+e^x \cos x$,
(d) $u^{\prime \prime}+u^{\prime}-2 u=\sinh x$, where $\sinh x=\frac{1}{2}\left(e^x-e^{-x}\right)$, (e) $u^{\prime \prime \prime}+9 u^{\prime}=1+e^{3 x}$.

Matthew Allcock

Numerade Educator

08:53

Problem 32

Dual Bases: Given a basis $\mathbf{v}_1, \ldots, \mathbf{v}_n$ of $V$, the dual basis $\ell_1, \ldots, \ell_n$ of $V^*$ consists of the linear functions uniquely defined by the requirements $\ell_i\left(\mathbf{v}_j\right)= \begin{cases}1 & i=j, \\ 0, & i \neq j .\end{cases}$
(a) Show that $\ell_i[\mathbf{v}]=x_i$ gives the $i^{\text {th }}$ coordinate of a vector $\mathbf{v}=x_1 \mathbf{v}_1+\cdots+x_n \mathbf{v}_n$ with respect to the given basis. (b) Prove that the dual basis is indeed a basis for the dual vector space. (c) Prove that if $V=\mathbb{R}^n$ and $A=\left(\mathbf{v}_1 \mathbf{v}_2 \ldots \mathbf{v}_n\right)$ is the $n \times n$ matrix whose columns are the basis vectors, then the rows of the inverse matrix $A^{-1}$ can be identified as the corresponding dual basis of $\left(\mathbb{R}^n\right)^*$.

Victor Salazar

Numerade Educator

Problem 33

Consider the differential equation $u^{\prime \prime}+x u=2$. Suppose you know solutions to the two boundary value problems $u(0)=1, u(1)=0$ and $u(0)=0, u(1)=1$. List all possible boundary value problems you can solve using superposition.

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11:27

Problem 33

Use Exercise 7.1.32(c) to find the dual basis for: (a) $\mathbf{v}_1=\left(\begin{array}{l}1 \\ 1\end{array}\right), \mathbf{v}_2=\left(\begin{array}{r}1 \\ -1\end{array}\right)$;
(b) $\mathbf{v}_1=\left(\begin{array}{l}1 \\ 2\end{array}\right), \mathbf{v}_2=\left(\begin{array}{r}3 \\ -1\end{array}\right)$;
(c) $\mathbf{v}_1=\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right), \mathbf{v}_2=\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right), \mathbf{v}_3=\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)$;
(d) $\mathbf{v}_1=$
$$
\left(\begin{array}{r}
1 \\
2 \\
-3
\end{array}\right), \mathbf{v}_2=\left(\begin{array}{r}
0 \\
-3 \\
1
\end{array}\right), \mathbf{v}_3=\left(\begin{array}{r}
-1 \\
2 \\
2
\end{array}\right) ;(e) \mathbf{v}_1=\left(\begin{array}{l}
1 \\
1 \\
0 \\
0
\end{array}\right), \mathbf{v}_2=\left(\begin{array}{l}
0 \\
1 \\
1 \\
0
\end{array}\right), \mathbf{v}_3=\left(\begin{array}{l}
0 \\
0 \\
1 \\
1
\end{array}\right), \mathbf{v}_4=\left(\begin{array}{r}
1 \\
-1 \\
1 \\
2
\end{array}\right) \text {. }
$$

Will Erickson

Numerade Educator

05:43

Problem 34

Consider the differential equation $x u^{\prime \prime}-(x+1) u^{\prime}+u=0$. Suppose we know the solution to the initial value problem $u(1)=2, u^{\prime}(1)=1$ is $u(x)=x+1$, while the solution to the initial value problem $u(1)=1, u^{\prime}(1)=1$ is $u(x)=e^{x-1}$. (a) What is the solution to the initial value problem $u(1)=3, u^{\prime}(1)=-2$ ? (b) What is the general solution to the differential equation?

Prakash Hampole

Numerade Educator

03:11

Problem 34

Let $\mathcal{P}^{(2)}$ denote the space of quadratic polynomials equipped with the $\mathrm{L}^2$ inner product $\langle p, q\rangle=\int_0^1 p(x) q(x) d x$. Find the polynomial $q$ that represents the following linear functions, i.e., such that $L[p]=\langle q, p\rangle$ :
(a) $L[p]=p(0)$,
(b) $L[p]=\frac{1}{2} p^{\prime}(1)$,
(c) $L[p]=\int_0^1 p(x) d x$,
(d) $L[p]=\int_{-1}^1 p(x) d x$.

Ahmad Reda

Numerade Educator

03:19

Problem 35

Consider the differential equation $4 x u^{\prime \prime}+2 u^{\prime}+u=0$. Given that $\cos \sqrt{x}$ solves the boundary value problem $u\left(\frac{1}{4} \pi^2\right)=0, u\left(\pi^2\right)=-1$, and $\sin \sqrt{x}$ solves the boundary value problem $u\left(\frac{1}{4} \pi^2\right)=1, u\left(\pi^2\right)=0$, write down the solution to the boundary value problem $u\left(4 \pi^2\right)=-3, u\left(\pi^2\right)=7$.

Madi Sousa

Numerade Educator

01:48

Problem 35

Find the dual basis, as defined in Exercise 7.1.32, for the monomial basis of $\mathcal{P}^{(2)}$ with respect to the $\mathrm{L}^2$ inner product $\langle p, q\rangle=\int_0^1 p(x) q(x) d x$.

Uma Kumari

Numerade Educator

02:53

Problem 36

Solve the following boundary value problems by using superposition: $(a) u^{\prime \prime}+9 u=x$,
$$
\begin{aligned}
& u(0)=1, u^{\prime}(\pi)=0, \text { (b) } u^{\prime \prime}-8 u^{\prime}+16 u=e^{4 x}, u(0)=1, u(1)=0, \quad \text { (c) } u^{\prime \prime}+4 u=\sin 3 x, \\
& u^{\prime}(0)=0, u(2 \pi)=3, \quad(d) u^{\prime \prime}-2 u^{\prime}+u=1+e^x, u^{\prime}(0)=-1, u^{\prime}(1)=1 .
\end{aligned}
$$

Srikar Pasumarthy

Numerade Educator

04:34

Problem 36

Write out a proof of Theorem 7.10 that does not rely on finding an orthonormal basis.

Chris Trentman

Numerade Educator

06:12

Problem 37

Given that $x^2+y^2$ solves the Poisson equation $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=4$, while $x^4+y^4$ solves $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=12\left(x^2+y^2\right)$, write down a solution to $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=1+x^2+y^2$.

Matthys Marthinus

Numerade Educator

06:39

Problem 37

For each of the following pairs of linear functions $S, T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, describe the compositions $S \circ T$ and $T \circ S$. Do the functions commute?
(a) $S=$ counterclockwise rotation by $60^{\circ} ; T=$ clockwise rotation by $120^{\circ}$;
(b) $S=$ reflection in the line $y=x ; T=$ rotation by $180^{\circ}$;
(c) $S=$ reflection in the $x$-axis; $T=$ reflection in the $y$-axis;
(d) $S=$ reflection in the line $y=x ; T=$ reflection in the line $y=2 x$;
(e) $S=$ orthogonal projection on the $x$-axis; $T=$ orthogonal projection on the $y$-axis;
(f) $S=$ orthogonal projection on the $x$-axis; $T=$ orthogonal projection on the line $y=x$;
(g) $S=$ orthogonal projection on the $x$-axis; $T=$ rotation by $180^{\circ}$;
(h) $S=$ orthogonal projection on the $x$-axis; $T=$ counterclockwise rotation by $90^{\circ}$;
(i) $S=$ orthogonal projection on the line $y=-2 x ; T=$ reflection in the line $y=x$.

Tamara Worner

Numerade Educator

00:43

Problem 38

Reduction of order: Suppose you know one solution $u_1(x)$ to the second order homogeneous differential equation $u^{\prime \prime}+a(x) u^{\prime}+b(x) u=0$. (a) Show that if $u(x)=$ $v(x) u_1(x)$ is any other solution, then $v(x)=v^{\prime}(x)$ satisfies a first order differential equation. (b) Use reduction of order to find the general solution to the following equations, based on the indicated solution:
(i) $u^{\prime \prime}-2 u^{\prime}+u=0, u_1(x)=e^x,(x) x u^{\prime \prime}+(x-1) u^{\prime}-u=0, u_1(x)=x-1$,
(iii) $u^{\prime \prime}+4 x u^{\prime}+\left(4 x^2+2\right) u=0, u_1(x)=e^{-x^2}$, (iv) $u^{\prime \prime}-\left(x^2+1\right) u=0, u_1(x)=e^{x^2 / 2}$.

Darshan Maheshwari

Numerade Educator

04:23

Problem 38

Find a matrix representative for the linear functions (a) $L: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ that maps $\mathbf{e}_1$ to $\left(\begin{array}{r}1 \\ -3\end{array}\right)$ and $\mathbf{e}_2$ to $\left(\begin{array}{r}-1 \\ 2\end{array}\right)$;
(b) $M: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ that takes $\mathbf{e}_1$ to $\left(\begin{array}{l}-1 \\ -3\end{array}\right)$ and $\mathbf{e}_2$ to $\left(\begin{array}{l}0 \\ 2\end{array}\right)$; and (c) $N: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ that takes $\left(\begin{array}{r}1 \\ -3\end{array}\right)$ to $\left(\begin{array}{l}-1 \\ -3\end{array}\right)$ and $\left(\begin{array}{r}-1 \\ 2\end{array}\right)$ to $\left(\begin{array}{l}0 \\ 2\end{array}\right)$.
(d) Explain why $M=N \circ L$. (e) Verify part (d) by multiplying the matrix representatives.

Elham Kordzadeh

Numerade Educator

Problem 39

Write out the details of the proof of Theorem 7.43 .

Check back soon!

11:28

Problem 39

On the vector space $\mathbb{R}^3$, let $R$ denote counterclockwise rotation around the $x$ axis by $90^{\circ}$ and $S$ counterclockwise rotation around the $z$-axis by $90^{\circ}$. (a) Find matrix representatives for $R$ and $S$. (b) Show that $R \circ S \neq S \circ R$. Explain what happens to the standard basis vectors under the two compositions. (c) Give an experimental demonstration of the noncommutativity of $R$ and $S$ by physically rotating a solid object, e.g., this book, in the prescribed manners.

Anthony Ramos

Numerade Educator

View

Problem 40

Can you find a complex matrix $A$ such that ker $A \neq\{0\}$ and the real and imaginary parts of every complex solution to $A \mathbf{u}=\mathbf{0}$ are also solutions?

Nick Johnson

Numerade Educator