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	\title{Weekly Assignment 3}
	
	\author{WRITE YOUR NAME HERE\\
		MATH 117: Advanced Linear Algebra}
	
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	Some hints for this assignment are written in the footnotes. See the \href{https://people.ucsc.edu/~jbreland/teaching/SM23_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 Let $V$ be finite-dimensional vector space and $W$ a subspace. Suppose that $\{b_1,\ldots,b_k\}$ is a basis for $W$ and extend this to a basis $\{b_1,\ldots,b_k,b_{k+1},\ldots,b_n\}$ for $V$ using \href{https://people.ucsc.edu/~jbreland/teaching/SM22_MATH117/complete_lecture_notes.html#b-1kUG8o6pTg}{Proposition 1.4.11}. Prove that the set of vectors $\{b_{k+1}+W,\ldots,b_n+W\}$ is a basis for the quotient space $V/W$. 
		
		\begin{proof} Write your proof here.
			
		\end{proof}
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	\end{enumerate}
	
	
	
	
	
	\begin{definition}Let $V$ be a vector space and let $S$ be a subset of $V$. The \emph{annihilator} $S^0$ of $S$ is the set of linear functionals whose kernel contains $S$, that is,
		\[
		S^0 := \{f\in V^* : f(v) = 0 \text{ for all }v\in S \} \subset V^*.
		\]
	\end{definition}
	
	
	\begin{enumerate}[\bfseries 1.]
		\setcounter{enumi}{1}
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		\item Let $V$ be a vector space. 
		
		\begin{enumerate}[(a)]
			
			\item Let $S$ be a subset of $V$. Prove that $S^0$ is a subspace of $V^*$.
			
			\item Let $W$ be a subspace of $V$. Prove that $W^0$ is isomorphic to $(V/W)^*$.\footnote{Hint: Universal Property of the Quotient.}
			
			\item Suppose that $V$ is finite-dimensional and let $W$ be a subspace of $V$. By part (b), $\dim(W^0) = \dim V - \dim W$. Provide another proof of this equation using dual bases.\footnote{Hint: Start with a basis for $W$ and extend to a basis for $V$. Can you use the corresponding dual basis to construct a basis for $W^0$?}
			
		\end{enumerate}
		
		\begin{proof} Write your proof here.
			
		\end{proof}
		
	\end{enumerate}
	
\begin{definition}Let $V$ be a vector space. A bilinear form $B:V\times V \to F$ is called \emph{reflexive} if $B(v,v') = 0$ implies $B(v',v) = 0$ for all $v,v'\in V$. The \emph{radical} of a reflexive bilinear form is the set
	\[
	\rad(V) : = \{v \in V : B(v,v') = 0 \text{ for all } v'\in V  \}.
	\]
	A reflexive bilinear form is called \emph{nondegenerate} if $\rad(V) = \{0\}$. 
\end{definition}


\begin{enumerate}[\bfseries 1.]
	\setcounter{enumi}{2}
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	\item Let $V$ be a vector space. Let $B:V\times V \to F$ be a bilinear form on $V$.
	
	\begin{enumerate}[(a)]
		\item  For any $v \in V$, define a function $\Phi_B(v):V \to F$ by the rule $(\Phi_B(v))(w) = B(v,w)$. Show that $\Phi_B(v)$ is a linear functional and show that the assignment $v \mapsto \Phi_B(v)$ defines a linear map $\Phi_B:V\to V^*$.		
		
		\item Suppose that $V$ is finite-dimensional and that $B$ is reflexive and nondegenerate. 
		\begin{enumerate}[(i)]
			\item Prove that $\Phi_B$ is an isomorphism. 
			
			\item Let $W$ be a subspace of $V$. Describe the preimage $W^\perp:=\Phi_B^{-1}(W^0)$ of $W^0$ under $\Phi_B$. In particular, $W^0 \cong W^\perp$. 
			
			\item Suppose that $B$ is nondegenerate when restricted to $W$, i.e., $\rad(W) = \{0\}$. Prove that $V = W\oplus W^\perp$.
		\end{enumerate}
		
		
	\end{enumerate}
	
	\begin{proof} Write your proof here.
		
	\end{proof}
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	\item \begin{enumerate}[(i)]
		
		\item	Suppose that $L_1:V_1\to W_1$ and $L_2: V_2 \to W_2$ are linear maps. Prove that there is a unique linear map
		\[
		L_1\otimes L_2 :V_1\otimes V_2 \to W_1 \otimes W_2
		\]
		with the property that $(L_1\otimes L_2)(v_1\otimes v_2) = L(v_1) \otimes L(v_2)$ for all $v_1 \in V_1$ and $v_2 \in V_2$.\footnote{Hint: Universal Property of the Tensor Product.}
		
		\item Let $F = \Z_5$ and let $V = F^2$. Let $L:V\to V$ be the linear map defined by left multiplication with the matrix $A = \begin{psmallmatrix}
			0 & 1\\ 4 & 2
		\end{psmallmatrix}$. Let $E=(e_1,e_2)$ denote the standard basis for $V$. Compute the matrix 
		\[
		[L\otimes L]_{B}	
		\]
		for linear map $L\otimes L : V\otimes V \to V \otimes V$, where $B$ is the basis $(e_1\otimes e_1, e_1\otimes e_2, e_2\otimes e_1, e_2\otimes e_2)$ for $V\otimes V$. 
	\end{enumerate}
	\begin{proof} Write your proof here.
		
	\end{proof}
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