Skip to main content
$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 6.2.E: Problems on Linear Maps and Matrices

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

Exercise $$\PageIndex{1}$$

Verify Note 1 and the equivalence of the two statements in Definition 1.

Exercise $$\PageIndex{2}$$

In Examples (b) and (c) show that
$f_{n} \rightarrow f(\text {uniformly}) \text { on } I \text { iff }\left\|f_{n}-f\right\| \rightarrow 0,$
i.e., $$f_{n} \rightarrow f$$ in $$E^{\prime}.$$
[Hint: Use Theorem 1 in Chapter 4, §2.]
Hence deduce the following.
(i) If $$E$$ is complete, then the map $$\phi$$ in Example (c) is continuous.
[Hint: Use Theorem 2 of Chapter 5, §9, and Theorem 1 in Chapter 4, §12.]
(ii) The map $$D$$ of Example (b}) is not continuous.
[Hint: Use Problem 3 in Chapter 5, §9.]

Exercise $$\PageIndex{3}$$

Prove Corollaries 1 to 3.

Exercise $$\PageIndex{3'}$$

Show that
$\|f\|=\sup _{|\vec{x}| \leq 1}|f(\vec{x})|=\sup _{|\vec{x}|=1}|f(\vec{x})|=\sup _{\vec{x} \neq \overrightarrow{0}} \frac{|f(\vec{x})|}{|\vec{x}|}.$
[Hint: From linearity of $$f$$ deduce that $$|f(\vec{x})| \geq|f(c x)|$$ if $$|c|<1.$$ Hence one may disregard vectors of length $$<1$$ when computing sup $$|f(\vec{x})|.$$ Why?]

Exercise $$\PageIndex{4}$$

Find the matrices $$[f],[g],[h],[k],$$ and the defining formulas for the linear maps $$f : E^{2} \rightarrow E^{1}, g : E^{3} \rightarrow E^{4}, h : E^{4} \rightarrow E^{2}, k : E^{1} \rightarrow E^{3}$$ if
(i) $$f\left(\vec{e}_{1}\right)=3, f\left(\vec{e}_{2}\right)=-2;$$
(ii) $$g\left(\vec{e}_{1}\right)=(1,0,-2,4), g\left(\vec{e}_{2}\right)=(0,2,-1,1), g\left(\vec{e}_{3}\right)=(0,1,0,-1);$$
(iii) $$h\left(\vec{e}_{1}\right)=(2,2), h\left(\vec{e}_{2}\right)=(0,-2), h\left(\vec{e}_{3}\right)=(1,0), h\left(\vec{e}_{4}\right)=(-1,1);$$
(iv) $$k(1)=(0,1,-1).$$

Exercise $$\PageIndex{5}$$

In Problem 4, use Note 4 to find the product matrices $$[k][f],[g][k],[f][h],$$ and $$[h][g].$$ Hence obtain the defining formulas for $$k \circ f, g \circ k, f \circ h,$$ and $$h \circ g.$$

Exercise $$\PageIndex{6}$$

For $$m \times n$$ matrices (with $$m$$ and $$n$$ fixed) define addition and multiplication by scalars as follows:
$a[f]+b[g]=[a f+b g] \text { if } f, g \in L\left(E^{n}, E^{m}\right)\left(\text { or } L\left(C^{n}, C^{m}\right)\right).$
Show that these matrices form a vector space over $$E^{1}$$ (or $$C$$).

Exercise $$\PageIndex{7}$$

With matrix addition as in Problem 6, and multiplication as in Note 4, show that all $$n \times n$$ matrices form a noncommutative ring with unity, i.e., satisfy the field axioms (Chapter 2, §§1-4) except the commutativity of multiplication and existence of multiplicative inverses (give counterex-amplest!).
Which is the "unity" matrix?

Exercise $$\PageIndex{8}$$

Let $$f : E^{\prime} \rightarrow E$$ be linear. Prove the following statements.
(i) The derivative $$D_{\vec{u}} f(\vec{p})$$ exists and equals $$f(\vec{u})$$ for every $$\vec{p}, \vec{u} \in E^{\prime} (\vec{u} \neq \overrightarrow{0});$$
(ii) $$f$$ is relatively continuous on any line in $$E^{\prime}$$ (use Theorem 1 in §1);
(iii) $$f$$ carries any such line into a line in $$E.$$

Exercise $$\PageIndex{9}$$

Let $$g : E^{\prime \prime} \rightarrow E$$ be linear. Prove that if some $$f : E^{\prime} \rightarrow E^{\prime \prime}$$ has a $$\vec{u}$$-directed derivative at $$\vec{p} \in E^{\prime},$$ so has $$h=g \circ f,$$ and $$D_{\vec{u}} h(\vec{p})=g\left(D_{\vec{u}} f(\vec{p})\right)$$.
[Hint: Use Problem 8.]

Exercise $$\PageIndex{10}$$

A set $$A$$ in a vector space $$V(A \subseteq V)$$ is said to be linear (or a linear subspace of $$V$$) iff $$a \vec{x}+b \vec{y} \in A$$ for any $$\vec{x}, \vec{y} \in A$$ and any scalars $$a, b.$$ Prove the following.
(i) Any such $$A$$ is itself a vector space.
(ii) If $$f : E^{\prime} \rightarrow E$$ is a linear map and $$A$$ is linear in $$E^{\prime}$$ (respectively, in $$E$$), so is $$f[A]$$ in $$E$$ (respectively, so is $$f^{-1}[A]$$ in $$E^{\prime}$$).

Exercise $$\PageIndex{11}$$

A set $$A$$ in a vector space $$V$$ is called the span of a set $$B \subseteq A(A=\operatorname{sp}(B))$$ iff $$A$$ consists of all linear combinations of vectors from $$B$$. We then also say that $$B$$ spans $$A$$.
Prove the following:
(i) $$A=\operatorname{sp}(B)$$ is the smallest linear subspace of $$V$$ that contains $$B$$.
(ii) If $$f : V \rightarrow E$$ is linear and $$A=\operatorname{sp}(B),$$ then $$f[A]=\operatorname{sp}(f[B])$$ in $$E$$.

Exercise $$\PageIndex{12}$$

A set $$B=\left\{\vec{x}_{1}, \vec{x}_{2}, \ldots, \vec{x}_{n}\right\}$$ in a vector space $$V$$ is called a basis iff each $$\vec{v} \in V$$ has a unique representation as
$\vec{v}=\sum_{i=1}^{n} a_{i} \vec{x}_{i}$
for some scalars $$a_{i}.$$ If so, the number $$n$$ of the vectors in $$B$$ is called the dimension of $$V,$$ and $$V$$ is said to be $$n$$- dimensional. Examples of such spaces are $$E^{n}$$ and $$C^{n}$$ (the $$\vec{e}_{k}$$ form a basis!).
(i) Show that $$B$$ is a basis iff it spans $$V$$ (see Problem 11) and its elements $$\vec{x}_{i}$$ are linearly independent, i.e.,
$\sum_{i=1}^{n} a_{i} \vec{x}_{i}=\overrightarrow{0} \text { iff all } a_{i} \text { vanish.}$
(ii) If $$E^{\prime}$$ is finite-dimensional, all linear maps on $$E^{\prime}$$ are uniformly continuous. (See also Problems 3 and 4 of §6.)

Exercise $$\PageIndex{13}$$

Prove that if $$f : E^{1} \rightarrow E$$ is continuous and $$\left(\forall x, y \in E^{1}\right)$$
$f(x+y)=f(x)+f(y),$
then $$f$$ is linear; so, by Corollary 2, $$f(x)=v x$$ where $$v=f(1)$$.
[Hint: Show that $$f(a x)=a f(x);$$ first for $$a=1,2, \ldots$$ (note: $$n x=x+x+\cdots+x, n$$ terms); then for rational $$a=m / n;$$ then for $$a=0$$ and $$a=-1.$$ Any $$a \in E^{1}$$ is a limit of rationals; so use continuity and Theorem 1 in Chapter 4, §2.]