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\begin{document}

\title{Math 101: Graduate Linear Algebra \\ Weekly Homework \#1}
\date{Assigned Friday, 15 September 2017; due Friday, 22 September 2017}

\maketitle

\begin{problab}{W1.1}
Let $V$ be a finite-dimensional vector space over a field $F$, and let $\phi\colon V\to V$ be an $F$-linear endomorphism of $V$.
  \begin{enumalph}
  \item Show that there exists $m \geq 0$ so that
    $\img(\phi^m) \cap \ker(\phi^m) = \{0\}$.
  \item Now suppose that $\phi^2 = 0$.  Show that the rank of
    $\phi$ is at most $(\dim V)/2$, and that there is a(n ordered) basis $\beta$ for $V$ such that $[T]_{\beta}$ has the block form
\[ \begin{pmatrix}
O & A \\
O & O 
\end{pmatrix} \]
i.e., has zeros in all blocks except possibly the upper right-hand corner.
\end{enumalph}
\end{problab}

\begin{problab}{W1.2}
Let $V,W$ be finite-dimensional vector spaces over a field $F$, and let $\phi\colon V \to W$ be an $F$-linear map.  
\begin{enumalph}
\item Show that there exists a basis $\beta$ of $V$ and a basis $\gamma$ of $W$ such that $[\phi]_{\beta}^{\gamma}$ is diagonal with diagonal entries in $\{0,1\}$.  What does the number of $1$s along the diagonal tell you about $\phi$?
\item Suppose now that $W=V$.  Recalling the result from daily homework, show that there exists a basis $\beta$ of $V$ such that the conclusion of (a) holds for $[\phi]_{\beta}$ if and only if $\phi^2=\phi$.
\end{enumalph} 
\end{problab}

\begin{problab}{W1.3}
Suppose $\opchar F \neq 2$.  Let $V$ be an $F$-vector space.  Recall that $\phi \in \End_F(V)$ is a projection if $\phi^2=\phi$.  

Let $\phi,\psi:V \to V$ be projection maps. 
\begin{enumalph}
\item Show that $\phi+\psi$ is a projection if and only if $\phi\psi=\psi\phi=0$ if and only if $\img \phi \subseteq \ker \psi$ and $\img \psi \subseteq \ker \phi$.
\item If $\phi+\psi$ is a projection, show that $\img(\phi+\psi)=\img(\phi)\oplus \img(\psi)$ and $\ker(\phi+\psi)=\ker(\phi)\cap\ker(\psi)$.
\end{enumalph}
\end{problab}

\begin{problab}{W1.4}
Let $V$ be an $F$-vector space with $n=\dim_F V<\infty$.  Let $A,B \subseteq V$ be $F$-subspaces with $a=\dim A$ and $b=\dim B$ and suppose $V=A+B$.  Let 
\[ S=\{f \in \End_F(V) : f(A) \subseteq A,\ f(B) \subseteq B\}. \]
Observe that $S \subseteq \End_F(V)$ is an $F$-subspace, and then express $\dim S$ in terms of $n,a,b$.
\end{problab}

\begin{problab}{W1.5}
Let $V,W$ be finite-dimensional $F$-vector spaces, let $X \subseteq W$ be an $F$-subspace, and let $\phi:V \to W$ be $F$-linear.  Prove that 
\[ \dim \phi^{-1}(X) \geq \dim V - \dim W + \dim X. \]
\end{problab}

\end{document}