6.2.E: Problems on Linear Maps and Matrices
Verify Note 1 and the equivalence of the two statements in Definition 1.
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.]
Prove Corollaries 1 to 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?]
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) \(k(1)=(0,1,-1).\)
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 \(h \circ g.\)
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 \(C\)).
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?
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 \(E.\)
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.]
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 \(E^{\prime}\)).
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 \(E\).
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.)
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.]