Linear Algebra in Action

Harry Dym

Chapter 7 Normed linear spaces - all with Video Answers

Educators

Chapter Questions

Problem 1

Show that if $\alpha, \beta \in \mathbb{R}$ and $\theta \in[0,2 \pi)$, then $\alpha \cos \theta+\beta \sin \theta \leq$ $\sqrt{\alpha^2+\beta^2}$ and that the upper bound is achieved for some choice of $\theta$.

Tanishq Gupta

Numerade Educator

04:03

Problem 2

Let $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ be nonnegative numbers and let $1<s<\infty$. Show that
$$
\left\{\sum_{k=1}^n\left|a_k+b_k\right|^s\right\}^{1 / s}=\left\{\sum_{k=1}^n\left|a_k\right|^s\right\}^{1 / s}+\left\{\sum_{k=1}^n\left|b_k\right|^s\right\}^{1 / s}
$$
if and only if the vectors $\mathbf{a}$ and $\mathbf{b}$ with components $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$, respectively, are linearly dependent. [HINT: See how to change the inequalities in the proof of Minkowski's inequality to equalities.]

Victor Salazar

Numerade Educator

06:26

Problem 3

Let $a_1, \ldots, a_n$ and $c_1, \ldots, c_n$ be positive numbers such that $c_1+\cdots+c_n=1$ and let $p>1$. Show that
$$
\left(\sum_{j=1}^n c_j a_j\right)^p \leq \sum_{j=1}^n c_j a_j^p
$$

Subham Jyoti Mishra

Numerade Educator

Problem 4

Verify Young's inequality when $n=3$ by exploiting the inequality (7.3) to show that if
$$
\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2} \quad \text { and } \quad \frac{1}{q}=\frac{1}{p_3},
$$
then:
(1) $a_1 a_2 a_3 \leq \frac{\left(a_1 a_2\right)^p}{p}+\frac{a_3^q}{q}$.
(2) $a_1 a_2 \leq \frac{p}{p_1} a_1^{p_1 / p}+\frac{p}{p_2} a_2^{p_2 / p}$.
(3) $\frac{p}{p_1} a_1^{p_1 / p}+\frac{p}{p_2} a_2^{p_2 / p} \leq\left(\frac{p}{p_1} a_1^{p_1}+\frac{p}{p_2} a_2^{p_2}\right)^{1 / p}$.
(4) Verify Young's inequality for $n=3$.

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Problem 5

Verify Young's inequality.

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03:14

Problem 6

Use Young's inequality to show that the geometric mean of a given set of positive numbers $b_1, \ldots, b_n$ is less than or equal to its arithmetic mean, i.e.,
$$
\left(b_1 b_2 \cdots b_n\right)^{1 / n} \leq \frac{b_1+b_2+\cdots+b_n}{n} .
$$

Jack Chen

Numerade Educator

Problem 7

Let $\mathcal{U}$ be a vector space over $\mathbb{C}$ with basis $\mathbf{u}_1, \ldots, \mathbf{u}_n$. Show that for each choice of $s$ in the interval $1 \leq s<\infty$ the formula
$$
\varphi\left(\sum_{j=1}^n x_j \mathbf{u}_j\right)=\left\{\sum_{j=1}^n\left|x_j\right|^s\right\}^{1 / s}
$$
also defines a norm on $\mathcal{U}$.

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Problem 8

Show that if $s \geq 1$ and $t \geq 0$, then
(7.13) $\quad\|\mathbf{x}\|_s \geq\|\mathbf{x}\|_{s+t} \geq\|\mathbf{x}\|_{\infty}$ for each vector $\mathbf{x} \in \mathbb{C}^n$.

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07:46

Problem 9

Show that $\lim _s \dagger_{\infty}\|\mathbf{x}\|_s=\|\mathbf{x}\|_{\infty}$ for each vector $\mathbf{x} \in \mathbb{C}^n$.

Shafiq Rehman

Numerade Educator

01:43

Problem 10

Sketch the sets $\left\{\mathbf{x} \in \mathbb{R}^2:\|\mathbf{x}\|_t \leq 1\right\}$ for $t=1,2$ and $\infty$.

Jason Orozco

Numerade Educator

Problem 11

Show that $\mathbb{C}^{p \times q}$ is a normed linear space over $\mathbb{C}$ with respect to each of the norms
$$
\|A\|_s=\left\{\begin{array}{l}
\left\{\sum_{i=1}^p \sum_{j=1}^q\left|a_{i j}\right|^s\right\}^{1 / s} \quad \text { if } 1 \leq s<\infty \\
\max \left\{\left|a_{i j}\right|: i=1, \ldots, p ; j=1, \ldots, q\right\} \quad \text { if } s=\infty,
\end{array}\right.
$$
in which $a_{i j}$ denotes the $i j$ entry of the matrix $A$.

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02:05

Problem 12

Show that if $A=\left[\begin{array}{ll}a & a \\ a & a\end{array}\right]$ with $a>0$, then $\left\|A^2\right\|_s>\|A\|_s^2$ when $2<s \leq \infty$.

Robert Daugherty

Numerade Educator

01:44

Problem 13

Show that the matrix $A$ defined in Exercise 7.12 satisfies the inequality if $4\left\|A^2\right\|_s>\|A\|_s^2$ for each choice of $s$ in the interval $1<s \leq$ $\infty$.
A subset $Q$ of a normed linear space $\mathcal{U}$ over $F$ is said to be convex if
$$
\mathbf{x}, \mathbf{y} \in Q \Longrightarrow t \mathbf{x}+(1-t) \mathbf{y} \in Q \text { for every } 0 \leq t \leq 1 \text {. }
$$

The balls of radius $r>0$
(7.15)
$$
B_r(\mathbf{a})=\{\mathbf{x} \in \mathcal{U}:\|\mathbf{a}-\mathbf{x}\|<r\} \text { and } \overline{B_r(\mathbf{a})}=\{\mathbf{x} \in \mathcal{U}:\|\mathbf{a}-\mathbf{x}\| \leq r\}
$$
are both convex sets.

Lucía Guerrero

Numerade Educator

Problem 14

Verify the claim that the open and closed balls defined in (7.15) are both convex.

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Problem 15

Let $A=\left[\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right]$ and $B=\left[\begin{array}{ll}0 & 1 \\ 0 & 1\end{array}\right]$. Calculate $\|A\|_s,\|B\|_s$, $\|A B\|_s$ and $\|B A\|_s$ for $1 \leq s \leq \infty$, and determine for which values of $s$, if any, $\|A B\|_s \leq\|A\|_s\|B\|_s$.

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Problem 16

Justify the formula for computing $\|A\|_{s, t}$ that is given in (3) of Lemma 7.11.

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Problem 17

Show that if $A=\left[\begin{array}{ll}a & b \\ 0 & c\end{array}\right] \in \mathbb{R}^{2 \times 2}$ and $d^2=a^2+b^2+c^2$, then
$$
\max \left\{\|A \mathbf{x}\|_2^2: \mathbf{x} \in \mathbb{R}^2 \text { and }\|\mathbf{x}\|_2=1\right\}=\frac{d^2+\sqrt{d^4-4 a^2 c^2}}{2} \text {. }
$$

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Problem 18

Compute the maximum eigenvalue of the matrix $A^H A$ when $A=\left[\begin{array}{ll}a & b \\ 0 & c\end{array}\right] \in \mathbb{R}^{2 \times 2}$ and show that it is equal to the maximum that was calculated in Exercise 7.17.

Victor Salazar

Numerade Educator

07:29

Problem 19

Verify the bound in (3) of Lemma 7.15.

Ernest Castorena

Numerade Educator

Problem 20

Let $A \in \mathbb{C}^{n \times n}$ and $\lambda \in \mathbb{C}$. Show that if $|\lambda|>\|A\|$, then $A$ is invertible and $\left\|\left(\lambda I_n-A\right)^{-1}\right\| \leq(|\lambda|-\|A\|)^{-1}$.

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Problem 21

Calculate $\|X\|, r_\sigma(X),\left(I_2-X\right)^{-1}$ and $\left\|\left(I_2-X\right)^{-1}\right\|$ for the matrix
$$
X=\left[\begin{array}{cc}
1 / 2 & 2 \\
0 & 1 / 2
\end{array}\right],
$$
using the formula in Exercise 7.17 to evaluate the norms.

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Problem 22

Show that if $X \in \mathbb{C}^{n \times n}$ and the spectral radius $r_\sigma(X)<1$, then $I_n-X$ is invertible and the sequence of matrices $\left\{S_k\right\}, k=0,1, \ldots$, defined in the proof of Lemma 7.15 is still a Cauchy sequence in $\mathbb{C}^{n \times n}$. However, the inequality $\left\|\left(I_n-X\right)^{-1}\right\| \leq\left(1-r_\sigma(X)\right)^{-1}$ may fail. [HINT: You may find Exercise 7.21 helpful.]

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02:06

Problem 23

Calculate $\max \left\{\left\|\left(I_3-X\right)^{-1} \mathbf{x}\right\|_2: \mathbf{x} \in \mathbb{R}^3\right.$ and $\left.\|\mathbf{x}\|_2=1\right\}$ for the matrix
$$
X=\left[\begin{array}{ccc}
1 / 2 & 1 & 0 \\
0 & 1 / 2 & 0 \\
0 & 0 & 3
\end{array}\right] .
$$

Runpeng Li

Numerade Educator

Problem 24

Let $f(\mathbf{x})$ be a linear functional on $\mathbb{C}^n$ and let $f\left(\mathbf{e}_j\right)=\alpha_j$ for $j=1, \ldots, n$, where $\mathbf{e}_j$ denotes the $j$ 'th column of $I_n$. Show that if $s>1$ and $t=s /(s-1)$, then
$$
\max \left\{|f(\mathbf{x})|:\|\mathbf{x}\|_s \leq 1\right\}=\left(\sum_{j=1}^n\left|\alpha_j\right|^t\right)^{1 / t} .
$$

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Problem 25

Let $f(\mathbf{x})$ be a linear functional on $\mathbb{C}^n$ and let $f\left(\mathbf{e}_j\right)=\alpha_j$ for $j=1, \ldots, n$, where $\mathbf{e}_j$ denotes the $j^{\prime}$ 'th column of $I_n$. Show that
$$
\max \left\{|f(\mathbf{x})|:\|\mathbf{x}\|_{\infty} \leq 1\right\}=\sum_{j=1}^n\left|\alpha_j\right| .
$$

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Problem 26

Let $f(\mathbf{x})$ be a linear functional on $\mathbb{C}^n$ and let $f\left(\mathbf{e}_j\right)=\alpha_j$ for $j=1, \ldots, n$, where $\mathbf{e}_j$ denotes the $j$ 'th column of $I_n$. Show that
$$
\max \left\{|f(\mathbf{x})|:\|\mathbf{x}\|_1 \leq 1\right\}=\max \left\{\left|\alpha_j\right|: j=1, \ldots, n\right\}
$$

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Problem 27

Show that if $p(\mathbf{x})$ is a seminorm on a vector space $\mathcal{X}$ over $\mathbb{F}$, then $p$ also automatically satisfies the following additional three conditions:
(3) $p(\mathbf{0})=0$.
(4) $p(\mathbf{x}) \geq 0$ for every $\mathbf{x} \in \mathcal{X}$.
(5) $p(\mathbf{x}-\mathbf{y}) \geq|p(\mathbf{x})-p(\mathbf{y})|$.

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05:45

Problem 28

Verify Theorem 7.20 for the case $\mathbb{F}=\mathbb{R}$.

Anthony Ramos

Numerade Educator

Problem 29

Show that if $\mathcal{X}$ is a finite dimensional normed linear space over $\mathbb{R}$, then Theorem 7.20 remains valid if $p(\mathbf{x})$ is only assumed to be a sublinear functional on $\mathcal{X}$; i.e., if for every choice of $\mathbf{x}, \mathbf{y} \in \mathcal{X}, p(\mathbf{x})$ satisfies the constraints (1) $\infty>p(\mathbf{x}) \geq 0$; (2) $p(\mathbf{x}+\mathbf{y}) \leq p(\mathbf{x})+p(\mathbf{y})$; (3) $p(\alpha \mathbf{x}=\alpha p(\mathbf{x})$ for all $\alpha>0$.

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Problem 30

Let $\mathcal{U}$ be the space of continuous real-valued functions $f(x)$ on the interval $0 \leq x \leq 1$ equipped with the norm
$$
\|f\|_u=\int_0^1|f(x)| d x .
$$

Show that $\mathcal{U}$ is not a Banach space.

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Problem 31

Let $\mathbf{u}_1, \ldots, \mathbf{u}_{\ell}$ be a basis for a normed linear space $\mathcal{U}$ over F. Show that the functional
$$
\varphi\left(\sum_{j=1}^{\ell} c_j \mathbf{u}_j\right)=\sum_{j=1}^{\ell}\left|c_j\right|
$$
defines a norm on $\mathcal{U}$.

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