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\fancyhead[L]{\textbf{MATH 510 Homework 2}}
\fancyhead[R]{\textbf{Your-Name-Goes-Here}}
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{\small
\begin{itemize}
\item Homework 2 is due February 2 at beginning of class.
\item Write the problem statement followed by a proof or solution.
\item List problems in the same order they were given.
\item If you skip a problem, include the problem statement with no solution. 
\end{itemize}}

\noindent\textbf{Corrected on Jan 28, 6:30pm: Problem 1(a) should say $\subseteq$, not $=$. In Problem 5, assume $n>1$ (or interpret $[0]^0=[1]$).}
\begin{enumerate}[{\rm 1.}]
\item Let $\mathbb{F}$ be an arbitrary field, and let $A_1,\dotsc,A_k \in \mathbb{F}^{m\times n}$ be matrices.
\begin{enumerate}[{\rm (a)}]
\item Show that $\operatorname{im}(A_1 + \cdots + A_k) \subseteq \operatorname{im} A_1 + \cdots + \operatorname{im} A_k$, as a sum of subspaces.
\item Show that $\operatorname{rank}(A_1 + \cdots + A_k) \leq \operatorname{rank} A_1 + \cdots + \operatorname{rank} A_k$.
\end{enumerate}

\smallskip

\item Let $V$ and $W$ be finite-dimensional vector spaces with ordered bases $\mathcal{B}=(b_1,b_2,\ldots,b_n)$ and $\mathcal{C}=(c_1,c_2,\ldots,c_m)$, respectively. Let $S:V\to V$ and $T:W\to W$ be linear maps. Define a function $S\oplus T:V\oplus W\to V\oplus W$ (external direct sum) by
\[(S\oplus T)(v,w)=\big(S(v),T(w)\big)\]
Let $\mathcal{D}=\big((b_1,0),\ldots,(b_n,0),\; (0,c_1),\ldots,(0,c_m)\big)$ be the corresponding ordered basis for $V\oplus W$.
\begin{enumerate}[{\rm (a)}]
\item Show that $S\oplus T$ is a linear map.
\item Show that the matrix $[S\oplus T]_{\mathcal{D}}$ is the direct sum of the matrices $[S]_{\mathcal{B}}$ and $[T]_{\mathcal{C}}$. (Recall that the direct sum of two matrices $A\in F^{n\times n}$ and $B\in F^{m\times m}$ is the $(n+m)\times (n+m)$-block matrix $\begin{bmatrix}A&0_{n,m}\\0_{m,n}&B\end{bmatrix}$ where $0_{a,b}\in F^{a\times b}$ denotes the zero matrix.)
\end{enumerate}

\smallskip

\item Let $X$ be any set and $\mathbb{F}$ be any field. Let $\mathbb{F}^X$ be the set of all functions from $X$ to $\mathbb{F}$ with pointwise operations $(f+g)(x)=f(x)+g(x)$ and $(\lambda f)(x)=\lambda f(x)$. For each $x\in X$, let $e_x\in \mathbb{F}^X$ be the characteristic function on $\{x\}$, defined by \[e_x(y)=\begin{cases}1,&\text{if $y=x$,}\\0,&\text{otherwise.}\end{cases}\] The \emph{free $\mathbb{F}$-vector space on $X$} is defined as $\mathbb{F}X=\operatorname{span}_\mathbb{F}\{e_x\mid x\in X\}$.
\begin{enumerate}[{\rm (a)}]
\item Show that $\{e_x\mid x\in X\}$ is a basis for $\mathbb{F}X$.
\item Show that for any function $\varphi:X\to W$ from $X$ into some vector space $W$, there is a unique linear map $\Phi:\mathbb{F}X\to W$ such that $\Phi(e_x)=\varphi(x)$ for all $x\in X$.
\item Let $X=U$ where $U$ is some vector space. Explain why, in $\mathbb{F}U$, it is the case that for all $u,v\in U$: $e_{u+v}\neq e_u + e_v$. Similarly, for $1\neq \lambda\in\mathbb{F}$ and $u\in U$, explain why $e_{\lambda u}\neq \lambda e_u$.
\end{enumerate}

\smallskip

\item Let $V,W$ be finite-dimensional vector spaces with ordered bases $\mathcal{B},\mathcal{C}$ respectively. Show that a linear transformation $T:V\to W$ is invertible if and only if its matrix $[T]_{\mathcal{B},\mathcal{C}}$ is invertible.

\smallskip

\item Let $V$ be a finite-dimensional vector space of dimension $n>1$ and suppose $T:V\to V$ is a linear map such that $T^n=0$ but $T^{n-1}\neq 0$. (Here $T^n=T\circ T\circ\cdots \circ T$ and $0$ means the zero in $\operatorname{End}(V)$.) Show that there is a basis $\mathcal{B}$ for $V$ such that 
\[[T]_{\mathcal{B}}=\sum_{i=1}^{n-1} E_{i+1,i}=
\begin{bmatrix}
0&\cdots&0&0\\
1&\cdots&0&0\\
\vdots&\ddots&0&0\\
0&\cdots&1&0
\end{bmatrix}.
\]
\end{enumerate}


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