• Home
  • Textbooks
  • Linear Algebra Done Right
  • Inner-Product Spaces

Linear Algebra Done Right

Sheldon Axler

Chapter 6

Inner-Product Spaces - all with Video Answers

Educators

Chapter Questions

04:44

Problem 1

Prove that if $x, y$ are nonzero vectors in $\mathbf{R}^{2}$, then
$$\langle x, y\rangle=\|x\|\|y\| \cos \theta$$
where $\theta$ is the angle between $x$ and $y$ (thinking of $x$ and $y$ as arrows with initial point at the origin). Hint: draw the triangle formed by $x, y,$ and $x-y ;$ then use the law of cosines.

Shafiq Rehman
Shafiq Rehman
Numerade Educator
07:25

Problem 2

Suppose $u, v \in V$. Prove that $\langle u, v\rangle=0$ if and only if
$$\|u\| \leq\|u+a v\|$$
for all $a \in \mathbf{F}$

Chris Trentman
Chris Trentman
Numerade Educator
08:48

Problem 3

Prove that
$$\left(\sum_{j=1}^{n} a_{j} b_{j}\right)^{2} \leq\left(\sum_{j=1}^{n} j a_{j}^{2}\right)\left(\sum_{j=1}^{n} \frac{b_{j}^{2}}{j}\right)$$
for all real numbers $a_{1}, \ldots, a_{n}$ and $b_{1}, \ldots, b_{n}$

Chris Trentman
Chris Trentman
Numerade Educator
01:46

Problem 4

Suppose $u, v \in V$ are such that
$$\|u\|=3, \quad\|u+v\|=4, \quad\|u-v\|=6$$
What number must $\|v\|$ equal?

Subhakanta Sahoo
Subhakanta Sahoo
Numerade Educator
View

Problem 5

Prove or disprove: there is an inner product on $\mathbf{R}^{2}$ such that the associated norm is given by
$$\left\|\left(x_{1}, x_{2}\right)\right\|=\left|x_{1}\right|+\left|x_{2}\right|$$
for all $\left(x_{1}, x_{2}\right) \in \mathbf{R}^{2}$

Victor Salazar
Victor Salazar
Numerade Educator
15:25

Problem 6

Prove that if $V$ is a real inner-product space, then
$$\langle u, v\rangle=\frac{\|u+v\|^{2}-\|u-v\|^{2}}{4}$$
for all $u, v \in V$

Donald Albin
Donald Albin
Numerade Educator
02:42

Problem 7

Prove that if $V$ is a complex inner-product space, then
$$\langle u, v\rangle=\frac{\|u+v\|^{2}-\|u-v\|^{2}+\|u+i v\|^{2} i-\|u-i v\|^{2} i}{4}$$
for all $u, v \in V$

Chris Trentman
Chris Trentman
Numerade Educator
03:46

Problem 8

A norm on a vector space $U$ is a function \|\|$: U \rightarrow[0, \infty)$ such that $\|u\|=0$ if and only if $u=0,\|\alpha u\|=|\alpha|\|u\|$ for all $\alpha \in \mathbf{F}$ and all $u \in U,$ and $\|u+v\| \leq\|u\|+\|v\|$ for all $u, v \in U .$ Prove that a norm satisfying the parallelogram equality comes from an inner product (in other words, show that if $\|$ If is a norm on $U$ satisfying the parallelogram equality, then there is an inner product $\langle,\rangle$ on $U$ such that $\|u\|=\langle u, u\rangle^{1 / 2}$ for all $u \in U$.

Chris Trentman
Chris Trentman
Numerade Educator
09:10

Problem 9

Suppose $n$ is a positive integer. Prove that
$$\left(\frac{1}{\sqrt{2 \pi}}, \frac{\sin x}{\sqrt{\pi}}, \frac{\sin 2 x}{\sqrt{\pi}}, \ldots, \frac{\sin n x}{\sqrt{\pi}}, \frac{\cos x}{\sqrt{\pi}}, \frac{\cos 2 x}{\sqrt{\pi}}, \ldots, \frac{\cos n x}{\sqrt{\pi}}\right)$$
is an orthonormal list of vectors in $C[-\pi, \pi]$, the vector space of continuous real-valued functions on $[-\pi, \pi]$ with inner product
$$(f, g)=\int_{-\pi}^{\pi} f(x) g(x) d x$$

Matthew Allcock
Matthew Allcock
Numerade Educator
09:54

Problem 10

On $P_{2}(\mathbf{R}),$ consider the inner product given by
$$\langle p, q\rangle=\int_{0}^{1} p(x) q(x) d x$$
Apply the Gram-Schmidt procedure to the basis (1, $x, x^{2}$ ) to produce an orthonormal basis of $\mathcal{P}_{2}(\mathbf{R})$

Anthony Ramos
Anthony Ramos
Numerade Educator
04:00

Problem 11

What happens if the Gram-Schmidt procedure is applied to a list of vectors that is not linearly independent?

Wendi Zhao
Wendi Zhao
Numerade Educator
02:10

Problem 12

Suppose $V$ is a real inner-product space and $\left(\nu_{1}, \ldots, \gamma_{m}\right)$ is a linearly independent list of vectors in $V$. Prove that there exist exactly $2^{m}$ orthonormal lists $\left(e_{1}, \ldots, e_{m}\right)$ of vectors in $V$ such that
$$\operatorname{span}\left(v_{1}, \ldots, v_{j}\right)=\operatorname{span}\left(e_{1}, \ldots, e_{j}\right)$$
for all $j \in\{1, \ldots, m\}$

Nick Johnson
Nick Johnson
Numerade Educator
10:45

Problem 13

Suppose $\left(e_{1}, \ldots, e_{m}\right)$ is an orthonormal list of vectors in $V$. Let $v \in V .$ Prove that
$$\|v\|^{2}=\left|\left\langle v, e_{1}\right\rangle\right|^{2}+\cdots+\left|\left\langle v, e_{m}\right\rangle\right|^{2}$$
for all $j \in\{1, \ldots, m\}$

Donald Albin
Donald Albin
Numerade Educator
View

Problem 14

Find an orthonormal basis of $\mathcal{P}_{2}(\mathrm{R})$ (with inner product as in Exercise 10 such that the differentiation operator (the operator that takes $p$ to $p^{\prime}$ ) on $P_{2}(\mathrm{R})$ has an upper-triangular matrix with respect to this basis.

Joseph David
Joseph David
Numerade Educator
01:32

Problem 15

Suppose $U$ is a subspace of $V$. Prove that
$$\operatorname{dim} U^{1}=\operatorname{dim} V-\operatorname{dim} U$$

Victor Salazar
Victor Salazar
Numerade Educator
01:25

Problem 16

Suppose $U$ is a subspace of $V$. Prove that $U^{\perp}=\{0\}$ if and only if $U=V$

ET
Ed Tam
Numerade Educator
04:01

Problem 17

Prove that if $P \in \mathcal{L}(V)$ is such that $P^{2}=P$ and every vector in null $P$ is orthogonal to every vector in range $P,$ then $P$ is an orthogonal projection.

Chris Trentman
Chris Trentman
Numerade Educator
04:01

Problem 18

Prove that if $P \in \mathcal{L}(V)$ is such that $P^{2}=P$ and
$$\|P v\| \leq\|v\|$$
for every $v \in V$, then $P$ is an orthogonal projection.

Chris Trentman
Chris Trentman
Numerade Educator
02:37

Problem 19

Suppose $T \in \mathcal{L}(V)$ and $U$ is a subspace of $V .$ Prove that $U$ is invariant under $T$ if and only if $P_{U} T P_{U}=T P_{U}$

ET
Ed Tam
Numerade Educator
02:37

Problem 20

Suppose $T \in \mathcal{L}(V)$ and $U$ is a subspace of $V .$ Prove that $U$ and $U^{+}$ are both invariant under $T$ if and only if $P_{U} T=T P_{U}$

ET
Ed Tam
Numerade Educator
02:16

Problem 21

$\ln \mathbf{R}^{4},$ let
$$U=\operatorname{span}((1,1,0,0),(1,1,1,2))$$
Find $u \in U$ such that $\|u-(1,2,3,4)\|$ is as small as possible.

Rukhmani Jain
Rukhmani Jain
Numerade Educator
04:41

Problem 22

Find $p \in \mathcal{P}_{3}(\mathrm{R})$ such that $p(0)=0, p^{\prime}(0)=0,$ and
$$\int_{0}^{1}|2+3 x-p(x)|^{2} d x$$
is as small as possible.

Ryo Kudo
Ryo Kudo
Numerade Educator
01:52

Problem 23

Find $p \in \mathcal{P}_{\mathrm{S}}(\mathbf{R})$ that makes
$$\int_{-\pi}^{\pi}|\sin x-p(x)|^{2} d x$$
as small as possible. (The polynomial 6.40 is an excellent approximation to the answer to this exercise, but here you are asked to find the exact solution, which involves powers of $\pi$. A computer that can perform symbolic integration will be useful.

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
05:06

Problem 24

Find a polynomial $q \in \mathcal{P}_{2}(\mathbf{R})$ such that
$$p\left(\frac{1}{2}\right)=\int_{0}^{1} p(x) q(x) d x$$
for every $p \in \mathcal{P}_{2}(\mathbf{R})$

Ahmad Reda
Ahmad Reda
Numerade Educator
00:52

Problem 25

Find a polynomial $q \in \mathcal{P}_{2}(\mathbf{R})$ such that
$$\int_{0}^{1} p(x)(\cos \pi x) d x=\int_{0}^{1} p(x) q(x) d x$$
for every $p \in \mathcal{P}_{2}(\mathbf{R})$

Ernest Castorena
Ernest Castorena
Numerade Educator
01:20

Problem 26

Fix a vector $v \in V$ and define $T \in \mathcal{L}(V, \mathbf{F})$ by $T u=\langle u, v\rangle .$ For $a \in \mathbf{F},$ find a formula for $T^{*} a$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:01

Problem 27

Suppose $n$ is a positive integer. Define $T \in \mathcal{L}\left(\mathbf{F}^{n}\right)$ by
$$T\left(z_{1}, \ldots, z_{n}\right)=\left(0, z_{1}, \ldots, z_{n-1}\right)$$
Find a formula for $T^{*}\left(z_{1}, \ldots, z_{n}\right)$

Vysakh M
Vysakh M
Numerade Educator
01:49

Problem 28

Suppose $T \in \mathcal{L}(V)$ and $\lambda \in \mathbf{F}$. Prove that $\lambda$ is an eigenvalue of $T$ if and only if $\bar{\lambda}$ is an eigenvalue of $T^{*}$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:37

Problem 29

Suppose $T \in \mathcal{L}(V)$ and $U$ is a subspace of $V .$ Prove that $U$ is invariant under $T$ if and only if $U^{\perp}$ is invariant under $T^{*}$

ET
Ed Tam
Numerade Educator
04:16

Problem 30

Suppose $T \in \mathcal{L}(V, W) .$ Prove that
(a) $\quad T$ is injective if and only if $T^{*}$ is surjective;
(b) $\quad T$ is surjective if and only if $T^{*}$ is injective.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
02:19

Problem 31

Prove that
$$\operatorname{dim} \text { null } T^{*}=\operatorname{dim} \text { null } T+\operatorname{dim} w-\operatorname{dim} v$$
and
$$\operatorname{dim} \text { range } T^{*}=\operatorname{dim} \operatorname{range} T$$
for every $T \in \mathcal{L}(V, W)$

Madi Sousa
Madi Sousa
Numerade Educator
01:28

Problem 32

Suppose $A$ is an $m$ -by- $n$ matrix of real numbers. Prove that the dimension of the span of the columns of $A$ (in $\mathbf{R}^{m}$ ) equals the dimension of the span of the rows of $A$ (in $\mathbf{R}^{n}$ ).

Harshita Goel
Harshita Goel
Numerade Educator