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		\title{Weekly Assignment 2}
		
		\author{WRITE YOUR NAME HERE\\
		MATH 117: Advanced Linear Algebra}
		
		\maketitle
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	Some hints for this assignment are written in the footnotes. 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 $V=U\oplus W$ is the direct sum of subspaces $U$ and $W$. The projection onto $U$ along $W$ is the function $P:V\to V$ defined via $P(v) = u$ where $u\in U$ is the unique vector satisfying $v = u + w$ for some $w\in W$. 
		\begin{enumerate}[(a)]			
			\item Prove that $P$ is linear.
			
			\item Prove that a linear operator $L:V\to V$ is a projection onto some subspace if and only if $L^2 = L$. 
		\end{enumerate}
		
		\begin{proof} Write your proof here.
	
		\end{proof}


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		\item Let $V_1,V_2$ be vector spaces over $F$. The direct sum $V_1\oplus V_2$ comes with linear maps 
		\[
		\iota_1:V_1 \to V_1\oplus V_2,\, v_1 \mapsto (v_1,0) \quad \text{and} \quad \iota_2:V_2\to V_1\oplus V_2, \, v_2 \mapsto (0,v_2).
		\]
	    Let $Z$ be any other vector space and let $L_1:V_1 \to Z$ and $L_2:V_2 \to Z$ be any other linear maps. Prove that there is a unique linear map $L:V_1\oplus V_2 \to Z$ with the property that $L\circ \iota_1 = L_1$ and $L\circ \iota_2 = L_2$. See Remark~\ref{rmk1}.
		
		\begin{proof} Write your proof here.
			
		\end{proof}
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		\item Let $V_1,V_2,W_1,W_2$ be vector spaces over $F$ and let $L_1:V_1\to W_1$ and $L_2:V_2\to W_2$ be linear maps. 
		\begin{enumerate}[(a)]
			\item Use the Universal Property of the Direct Sum (see Remark~\ref{rmk1}) to show that there is a unique linear map
			\[
			L_1\oplus L_2:V_1\oplus V_2 \to W_1\oplus W_2
			\]
			satisfying $(L_1\oplus L_2)(v_1,v_2) = (L_1(v_1),L_2(v_2))$. 
			
			\item Suppose additionally that $V_1,V_2,W_1,W_2$ are finite-dimensional with ordered bases $B_1=(a_1,\ldots,a_k)$, $B_2=(b_1,\ldots,b_l)$, $C_1 = (c_1,\ldots,c_m)$, and $C_2=(d_1,\ldots,d_n)$, respectively.
			\begin{enumerate}[(i)]
				\item Prove that $B:=((a_1,0),\ldots,(a_k,0),(0,b_1),\ldots,(0,b_l))$ is a basis for $V_1 \oplus V_2$. Similarly, $C:= ((c_1,0),\ldots,(c_l,0),(0,d_1),\ldots,(0,d_n))$ is a basis for $W_1\oplus W_2$.
				
				\item Prove that the matrix for $L_1\oplus L_2$ with respect to $B$ and $C$ has the following block diagonal form:
				\[
				[L_1\oplus L_2]_B^C = \begin{pmatrix}[L_1]_{B_1}^{C_1} & 0\\ 0 & [L_2]_{B_2}^{C_2}
				\end{pmatrix}.
				\]
			\end{enumerate}
		\end{enumerate}
		
		\begin{proof} Write your proof here.
			
		\end{proof}
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		\item Let $V$ be finite-dimensional vector space and $W$ a subspace. Suppose that $\{b_1,\ldots,b_n\}$ is a basis for $W$ and extend this to a basis $\{b_1,\ldots,b_k,b_{k+1},\ldots,n\}$ 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}


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\begin{remark}\label{rmk1} In other words, $L$ is the unique linear map making the following diagram commute
	% https://q.uiver.app/?q=WzAsNCxbMSwxLCJWXFxvcGx1cyBXIl0sWzAsMCwiViJdLFswLDIsIlciXSxbMywxLCJaIl0sWzEsMCwiXFxpb3RhX1YiLDJdLFsyLDAsIlxcaW90YV9XIl0sWzEsMywiTF9WIiwwLHsiY3VydmUiOi0yfV0sWzIsMywiTF9XIiwyLHsiY3VydmUiOjJ9XSxbMCwzLCJcXGV4aXN0cyAhIEwiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=
	\[\begin{tikzcd}
		V_1 \\
		& {V_1\oplus V_2} && Z \\
		V_2
		\arrow["{\iota_1}"', from=1-1, to=2-2]
		\arrow["{\iota_2}", from=3-1, to=2-2]
		\arrow["{L_1}", curve={height=-12pt}, from=1-1, to=2-4]
		\arrow["{L_1}"', curve={height=12pt}, from=3-1, to=2-4]
		\arrow["{\exists ! L}", dashed, from=2-2, to=2-4]
	\end{tikzcd}\]
	The property described in Problem 1 is usually called the Universal Property of the Direct Product. It provides a precise answer to the question ``How do I define a linear map out of the direct sum of two vector spaces?". One simply defines linear maps $L_1$ and $L_2$ as in the statement. Your proof will provide the recipe for constructing the desired linear map $L$. Moreover, it establishes a bijection of sets
	% https://q.uiver.app/?q=WzAsNCxbMCwwLCJIb20oVlxcb3BsdXMgVyxaKSJdLFsxLDAsIkhvbShWLFopXFxvcGx1cyBIb20oVyxaKSJdLFswLDEsIkwiXSxbMSwxLCIoTFxcY2lyY1xcaW90YV9WLExcXGNpcmNcXGlvdGFfVykiXSxbMCwxLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJhcnJvd2hlYWQifX19XSxbMiwzLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV1d
	\[\begin{tikzcd}
		{\Hom_F(V_1\oplus V_2,Z)} & {\Hom_F(V_1,Z)\oplus \Hom_F(V_2,Z)} \\
		L & {(L\circ\iota_1,L\circ\iota_2)}
		\arrow[tail reversed, from=1-1, to=1-2]
		\arrow[maps to, from=2-1, to=2-2]
	\end{tikzcd}\]
	However, notice that the domain and codomain are actually vector spaces. It can be shown that this bijection is a linear map, i.e., a vector space isomorphism. Note that all of this readily generalizes to the direct sum of finitely many vector spaces. Compare all of this with the discussion on the Universal Property of the Quotient Space from the lecture.
\end{remark}



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