\documentclass{amsart}
\usepackage{amssymb}
\newcommand{\F}{\mathbb F}
\newcommand{\L}{\mathcal L}

\begin{document}
\title{Problem Set \#1}
\author{Due September 8}
\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 $A\cdot adj(A)=\det(A)I_n=adj(A)\cdot A$.
\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 there is a natural isomorphism
\[
\L(W,U\oplus V)\to\L(W,U)\oplus L(W,V)
\]
given by sending $T$ to $\pi_U\circ T \oplus \pi_V\circ T$.
\end{enumerate}

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3.
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4.

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5.

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6.

\end{document} 