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	\title{Weekly Assignment 5}
	
	\author{WRITE YOUR NAME HERE\\
		MATH 117: Advanced Linear Algebra}
	
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	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. \bigskip
	
	
	For Problems 1 \& 2, $L:V\to V$ is a linear operator on a finite dimensional vector space $V$ over a field $F$ which contains the roots of the characteristic polynomial of $L$. For any positive integer $i\geq 1$ and eigenvalue $\lambda \in F$ for $L$, we set $V_i : = \ker((L-\lambda \Id_V)^i)$ and $\dim V_i = r_i$. The first two problems were lemmas used in the sketch of the proof of the Jordan basis theorem. 
	
	
	
	\begin{enumerate}[\bfseries 1.] 
		
		
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		\item \begin{enumerate}[(a)]
			\item Suppose that $i\geq 1$ is a positive such that $V_i = V_{i+1}$. Prove that $V_{(\lambda)} = V_i$. 
			
			\item In the lecture, we proved that the multiplicity $m$ of $\lambda$ in $m_L(x)$ is the smallest integer with the property that $V_{(\lambda)} = V_m$. Prove that the dimensions of the $V_i$ strictly increase until we reach $V_m$:
			\[
			\{0\} \subsetneq V_1 \subsetneq V_2 \subsetneq \cdots \subsetneq V_{m-1} \subsetneq V_m = V_{m+1} = \cdots
			\]
			This proves a method for computing $m$ - just compute $\dim(V_i)$ until the first time the dimension does not increase. That is, $m$ is the smallest $i$ such that $\dim(V_i) = \dim(V_{i+1})$. 
		\end{enumerate} 
		
		\begin{proof}
			Write your proof here.
		\end{proof}
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		\item Set $K = L-\lambda \Id_V$. Assume $i \geq 2$. Suppose that $v_1,\ldots,v_{r_i} \in V_i$ are chosen such that the cosets $v_1 +V_{i-1}, \ldots, V_{r_i} + V_{i-1}$ form a basis for $V_i/ V_{i-1}$. 
		\begin{enumerate}[(a)]
			\item Prove that $K(v_1),\ldots, K(v_{r_i}) \in V_{i-1}$.
			
			\item Prove that the cosets $K(v_1)+V_{i-2},\ldots, K(v_{r_i})+V_{i-2}$ are independent in $V_{i-1}/V_{i-2}$. 
		\end{enumerate}
		
		
		\begin{proof}
			Write your proof here.
		\end{proof}
		
		
		
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		\item Let $L:V\to V$ be a linear operator on a finite dimensional inner product space $V$. 
		\begin{enumerate}[(a)]
			\item Prove that there is a unique linear operator $L^*:V\to V$ satisfying $\langle L(x), y \rangle = \langle x L^*(y) \rangle$ for all $x,y\in V$. 
			
			\item Prove that a subspace $W$ of $V$ is $L$-invariant if and only if $W^\perp$ is $L^*$-invariant.
		\end{enumerate}	
			
		\begin{proof}
			Write your proof here.
		\end{proof}
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		\item Let $B$ be an ordered orthonormal basis for a finite dimensional inner product space $V$. Compute $[L^*]_B$. 
		 
		\begin{proof}
			Write your proof here.
		\end{proof}
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	\end{enumerate}
	
	
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