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	\title{Weekly Assignment 4}
	
	\author{WRITE YOUR NAME HERE\\
		MATH 117: Advanced Linear Algebra}
	
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	In this assignment, you will prove the Cayley-Hamilton theorem. Throughout, $V$ is a finite-dimensional vector space and $L:V\to V$ is a linear operator.  See the \href{https://people.ucsc.edu/~jbreland/teaching/SM22_MATH117/weekly_assignments.html}{weekly assignment webpage} for due dates, templates, and assignment description.
	Make sure to justify any claims you make. You may not appeal to any results that we have not discussed in class. 
	
	
	
	\begin{enumerate}[\bfseries 1.] 
		
		
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		\item Suppose that $W$ is an $L$-invariant\footnote{See Def 4.3.6} subspace of $V$. Then we obtain by restriction a linear operator $L\vert_W :W \to W$. Prove that the characteristic polynomial of $L\vert_W$ divides the characteristic polynomial of $L$.\footnote{Hint: start with a basis for $W$ and extend to a basis $B$ for $V$. Argue that $[L]_B = \begin{psmallmatrix}
				X & Y\\ 0 & Z
		\end{psmallmatrix}$ for some matrices $X,Y,Z$. Compute $c_L(x)$ using this fact. }
		
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		\item For $v\neq 0$, define $\Span(L,v):= \Span\{L^i(v) : i \in \N_0\}$. Here $L^0 : = \mathrm{Id}_V$. 
		\begin{enumerate}[(a)]
			\item Prove that $\Span(L,v)$ is an $L$-invariant subspace of $V$. 
			
			\item Prove that there exists a largest integer $k$ such that $B_k:=\{v,L(v),\ldots,L^{k-1}(v)\}$ is independent. Moreover, show that $B_k$ is a basis for $\Span(L,v)$.
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		\item Let $v \neq 0$ and let $B:=B_k$ be the basis for $W:= \Span(L,v)$ from Problems 1 \& 2. Define $m_{L,v}(x) = a_0 + a_1x + \dots +a_{k-1}x^{k-1} + x^k \in F[x]$ where $a_0,a_1,\ldots,a_{k-1}$ are the unique coefficients such that
		\[
		a_0v + a_1L(v) + \cdots+ a_{k-1}L^{k-1}(v) + L^k(v) = 0.
		\]
		Since $W$ is $L$-invariant, we obtain by restriction a linear operator $L\vert_W:W \to W$. The goal of this problem is to show that the characteristic polynomial of $L\vert_W$ is equal to $m_{L,v}(x)$. 
		
		\begin{enumerate}[(a)]
			\item Compute the matrix $xI_k - [L\vert_W]_B$.
			
			\item Show that $\det(xI_k - [L\vert_W]_B) = m_{L,v}(x)$ using induction on $k$. \footnote{Hint: For the inductive step, start by using cofactor expansion along the first row.}
		\end{enumerate}
		
		\begin{proof} Write your proof here.
			
		\end{proof}
		
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		\item Use Problems 1-3 to prove the Cayley-Hamilton theorem: the linear operator $L$ is a root of its characteristic polynomial, that is, $\mathrm{c}_L(L)$ is the zero operator in $\End(V)$.\footnote{Hint: need to show $c_L(L)(v) = 0$ for all $v\in V$. Case 1: $v=0$. Case 2: $v\neq 0$, invoke Problems 1 \& 3 to write $c_L(x)$ as a product of two polynomials. Use this to show $c_L(L)(v) = 0$.} 
		\begin{proof} Write your proof here.
			
		\end{proof}
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