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\begin{document}
\title{Problem Set \#2}
\author{Due October 6}
\maketitle

1. Basic Properties of the Determinant. In this problem, let $A$ be an $n\times n$ matrix over $\F$.
\begin{enumerate}
\item Using the definition, show that if $A$ has two rows which are scalar multiples of each other (so for some $i$ and $k$, $a_{i,j}=\lambda a_{k,j}$ for all $j$), then $|A|=0$.
\item Show that $|A|=|A^t|$.
\item Again using the definition, show that if we swap two rows or columns in a matrix, then we swap the sign of the determinant (you may use without proof the following facts: $sign(\sigma\circ\tau)=sign(\sigma)sign(\tau)$, and if $f_{i,j}$ is the bijection from $\{1,\dots,n\}$ to itself that swaps $i$ and $j$ and fixes everything else, then $sign(f_{i,j})=-1$.)
\item Show $A\cdot adj(A)=\det(A)I_n=adj(A)\cdot A$.
\item[5*] This problem is optional! Justify the results about the sign representation used in $3$. The basic idea is that anything can be written as a product of transpositions $f_{i,j}$ (and as a fun pop-culture reference, this group and these ideas occurred in a recent episode of Futurama).
\end{enumerate}

\bigskip

2. Let $U,V, W$ be vector spaces over $\F$. This problem and the following will explore the ``categorical'' properties of finite direct sums=finite direct products.
\begin{enumerate}
\item Let $\pi_U\colon U\oplus V\to U$ and $\pi_V\colon U\oplus V\to V$ be the projection maps $\pi_U(\vec{u}+\vec{v})=\vec{u}$ (resp for $V$). Show that these are surjective linear transformations with $\ker(\pi_V)=U$ and $\ker(\pi_U)=V$.
\item Show that we have a natural isomorphism
\[
\L(W,U\oplus V)\to\L(W,U)\oplus\L(W,V)
\]
given by $T\mapsto \pi_U\circ T\oplus\pi_V\circ T$.
\item Now let $i_U$ (resp $i_V$) denote the inclusion $U\to U\oplus V$. Show that these are injective linear transformations and $\pi_V\circ i_U=0$, $\pi_U\circ i_V=0$.
\item Show that we have a natural isomorphism
\[
\L(U\oplus V,W)\to \L(U,W)\oplus\L(V,W)
\]
given by $T\mapsto T\circ i_U\oplus T\circ i_V$.
\end{enumerate}

\bigskip

3. This problem will play with the notion of ``equal'' versus ``isomorphic''. It also stresses some of the pathologies with infinite dimensional spaces. Let $\R[x]$ be the vector space of all polynomials in $x$ over $\R$.
\begin{enumerate}
\item Show that $\R[x]$ is infinite dimensional.
\item Show that the subspace of all polyomials which have a root at $0$ is isomorphic to $\R[x]$. Conclude that if we just know $V\oplus W\cong V\oplus U$, then we can't conclude that $W\cong U$.
\item Show that the derivative is a surjective map $\R[x]\to\R[x]$. Compute the kernel. Conclude that a surjective map between infinite dimensional vector spaces need not be an isomorphism.
\item Show that the integral $\int_0^x$ provides a one-sided inverse to the derivative. Determine the image. Conclude that an injective map between infinite dimensional vector spaces need not be an isomorphism.
\item There is a standard basis for $\R[x]$: the basis of all monic monomials in $x$. If we mimic the arguments given in class, then we can associate an $\infty\times\infty$-matrix to any linear transformation (indexed by $\N\times\N$, so really it has a well-defined ``upper left corner''). Find the matrix for the derivative and integral.
\item Building on the previous part, does any $\infty\times\infty$-matrix (again, indexed by $\N\times\N$) occur as the matrix associated to a linear operator? What is the image of the map that sends a linear operator to its matrix?
\end{enumerate}


\end{document} 