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\begin{document}
\section*{Lior Silberman's Math 223: Problem Set 6 (due 1/3/2021)}
\begin{center}
\textbf{Practice problems (recommended, but do not submit)}
\par\end{center}
\begin{lyxlist}{10.}
\item [{A.}] Let $U,V,W,X$ be vector spaces.
\begin{lyxlist}{10.}
\item [{(a)}] Let $A\in\Hom(U,V)$, $B\in\Hom(W,X)$. We define maps $R_{A}\colon\Hom(V,W)\to\Hom(U,W)$,
$L_{B}\colon\Hom(V,W)\to\Hom(V,X)$ and $S_{A,B}\colon\Hom(V,W)\to\Hom(U,X)$
by $R_{A}(T)=TA$, $L_{B}(T)=BT$, $S_{A,B}(T)=BTA$. Show that all
three maps are linear.
\item [{(b)}] Suppose that $A,B\in\Hom(U,U)$ are invertible, with inverses
$A^{-1},B^{-1}$. Show that $AB$ is invertible, with inverse $B^{-1}A^{-1}$
(note the different order!)
\item [{(c)}] Let $A\in\Hom(U,V)$, $B\in\Hom(V,W)$. Show that $\Ker A\subset\Ker(BA)$
and that $\Image(BA)\subset\Image(B)$.
\item [{(d)}] Let $A\in\Hom(U,V)$, $B\in\Hom(V,W)$. If $BA$ is injective
then so is $A$. If $BA$ is surjective then so is $B$.
\end{lyxlist}
\item [{B.}] Let $X$ be a set, and let $M_{g}\colon\R^{X}\to\R^{X}$ be
the operator of multiplication by $g\in\R^{X}$. Show that $M_{g}$
is linear.
\end{lyxlist}
\begin{center}
\textbf{Isomorphism of vector spaces}
\par\end{center}
Let $U,V$ be two vector spaces.
\begin{lyxlist}{10.}
\item [{C.}] Fix a basis $B\subset U$.
\begin{lyxlist}{10.}
\item [{({*}a)}] Let $f\in\Hom(U,V)$ be a linear isomorphism. Show that
the image $f(B)=\left\{ f(\underline{v})\mid\underline{v}\in B\right\} $
is a basis of $V$.
\item [{RMK}] It follows that is $U$ is isomorphic to $V$ then $\dim U=\dim V$.
\item [{({*}{*}b)}] Conversely, suppose that $B'\subset V$ is a basis,
and and that $g\colon B\to B'$ is a function which is $1-1$ and
onto (see notations file). Show that there is an isomorphism $f\in\Hom(U,V)$
which agrees with $g$ on $B$.
\item [{RMK}] It follows that if $\dim U=\dim V$ then $U$ is isomorphic
to $V$.\bigskip{}
\end{lyxlist}
\item [{D.}] Let $T\in\Hom(U,V)$, $S\in\Hom(V,U)$. Show that the following
are equivalent
\begin{lyxlist}{10.}
\item [{(1)}] $ST=\Id_{V}$, $TS=\Id_{U}$.
\item [{(2)}] $S$ is invertible with inverse $T$.\bigskip{}
\end{lyxlist}
\item [{1.}] Suppose that $\dim U=\dim V<\infty$. Let $A\in\Hom(U,V)$.
Show that the following are equivalent:
\begin{lyxlist}{10.}
\item [{(1)}] $A$ is invertible.
\item [{(2)}] $A$ is surjective.
\item [{(3)}] $A$ is injective.\bigskip{}
\end{lyxlist}
\end{lyxlist}
\begin{center}
\textbf{Linear equations}
\par\end{center}
\begin{lyxlist}{10.}
\item [{2.}] (Recognition) Express the following equations as linear equations
by finding appropriate spaces, linear map, and constant vector.
\begin{lyxlist}{10.}
\item [{(a)}] $\begin{cases}
5x+7y & =3\\
z+2x & =1\\
2y+x+3z & =-1\\
x+y & =0
\end{cases}$.
\item [{(b)}] (Bessel equation) $x^{2}\frac{\diff^{2}y}{\dfx^{2}}+x\frac{\dfy}{\dfx}+(x^{2}-\alpha^{2})y=0$.
Use the space $\Ci(\R)$ of functions on $\R$ which can be differentiated
to all orders.
\item [{({*}c)}] Fixing $S,B\in\Hom(U,U)$ with $S$ invertible, $SXS^{-1}=B$
for an unknown $X\in\Hom(U,U)$PRAC. (Show that the map you define
is linear!)
\item [{PRAC}] Suppose that $\dim U=n$. Using a basis for $U$, replace
the equation of (c) with a system of $n^{2}$ equations in $n^{2}$unknowns.
\end{lyxlist}
\end{lyxlist}
\newpage{}
\begin{center}
\textbf{Similarity of matrices.}\bigskip{}
\par\end{center}
Let $U$ be a vector space. Write $\End(U)$ for $\Hom(U,U)$ (linear
maps from $U$ to itself). We develop here a crucial concept.
\begin{defn*}
We say that two transformations $A,B\in\End(U)$ are \emph{similar}
if there is an invertible linear map $S\in\End(U)$ such that $B=SAS^{-1}$.
\end{defn*}
\begin{lyxlist}{10.}
\item [{3.}] (Calculations)
\begin{lyxlist}{10.}
\item [{PRAC}] Suppose that $A,B$ are similar and $A=0$. Show that $B=0$.
\item [{(a)}] Suppose that $A,B$ are similar and $A=\Id_{U}$. Show that
$B=\Id_{U}$.
\item [{(b)}] Show that the matrices $A=\left(\begin{array}{cc}
0 & 2\\
6 & -4
\end{array}\right)$, $B=\left(\begin{array}{cc}
-33 & 15\\
-63 & 29
\end{array}\right)$ are similar via the similarity transformation $S=\left(\begin{array}{cc}
1 & 2\\
3 & 4
\end{array}\right)$. (For a formula for $S^{-1}$ see PS5)\bigskip{}
\end{lyxlist}
\item [{4.}] (Meaning of similarity) Let $\cB=\left\{ \vv_{i}\right\} _{i\in I}\subset U$
be a basis. By a practice problem C above, $\cB'=\left\{ S\vv_{i}\right\} _{i\in I}\subset U$
is also a basis. Let $M\in M_{I}(\R)$ be the matrix of $A$ with
respet to the basis $\cB$. Show that $M$ is also the matrix of $B=SAS^{-1}$
with respect to the basis $\cB'$.
\begin{lyxlist}{10.}
\item [{RMK}] We'll later show that similarity as another, different meaning:
similar matrices represent \emph{the same} transformation with respect
to \emph{different} bases. \bigskip{}
\end{lyxlist}
\item [{5.}] (Similarity is an ``equivalence relation'')
\begin{lyxlist}{10.}
\item [{(a)}] show that $A$ is similar to $A$ for all $A$. (Hint: choose
$S$ wisely)
\item [{(b)}] Suppose that $A$ is similar to $B$. Show that $B$ is similar
to $A$ (Hint: solve $B=SAS^{-1}$ for $A$).
\item [{(c)}] Suppose that $A$ is similar to $B$, and $B$ is simlar
to $C$. Show that $A$ is similar to $C$.\bigskip{}
\end{lyxlist}
\end{lyxlist}
For the rest of the problem set fix $A,B,S$ such that $B=SAS^{-1}$.
Define $A^{n}$ as follows: $A^{0}=\Id_{U}$ and $A^{n+1}=A^{n}\cdot A$.
\begin{lyxlist}{10.}
\item [{6.}] (Induction practice 1)
\begin{lyxlist}{10.}
\item [{(a)}] Show that $B^{0}=SA^{0}S^{-1}$
\item [{PRAC}] Show that $B^{2}=SA^{2}S^{-1}$ and $B^{3}=SA^{3}S^{-1}$.
\item [{(b)}] Suppose that $B^{n}=SA^{n}S^{-1}$. Show that $B^{n+1}=SA^{n+1}S^{-1}$.
\item [{~}] The principle of mathematical induction says that (a),(b)
together show that $B^{n}=SA^{n}S^{-1}$ for all $n$.\bigskip{}
\end{lyxlist}
\item [{SUPP}] (Induction practice 2) For a polynomials $p(x)=\sum_{i=0}^{n}a_{i}x^{i}\in\R[x]$
and $A\in\End(U)$ define $p(A)=\sum_{i=0}^{n}a_{i}A^{i}$. We will
prove that $p(B)=Sp(A)S^{-1}$.
\begin{lyxlist}{10.}
\item [{(a)}] Suppose that $p$ is a constant polynomial. Show that $p(B)=Sp(A)S^{-1}$.
\item [{(b)}] Suppose that the formula holds for polynomials of degree
at most $n$. Show that the formula holds for polynomials of degree
at most $n+1$ (hint: if $p$ has degree at most $n+1$ you can write
it as $p(x)=a_{n+1}x^{n+1}+q(x)$ where $q$ has degree at most $n$).
\item [{RMK}] You will need to show that $S(aT)S^{-1}=aSTS^{-1}$ for any
scalar $a$.
\item [{(c)}] Let $q(x)=\sum_{j=0}^{m}b_{j}x^{j}\in\R[x]$ be another polynomial,
and let $r(x)=p(x)q(x)$ their product in $\R[x]$. Show that $r(A)=p(A)q(A)$.
\item [{RMK}] Part (c) seems silly, but checking that things work ``the
way they are supposed to'' is important. To understand the motivation
note that we think of polynomial as \emph{formal expressions} rather
than functions -- we need to make a \emph{definition} to interpret
them as functions, and then we need to verify that this definition
works as expected.
\end{lyxlist}
\end{lyxlist}
\end{document}