%%---------------------------------------------------------------%%
%%-----------------------Packages-----------------------%%
%%---------------------------------------------------------------%%
%Some packages are essential for compiling your document!
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amsthm,amssymb}
\usepackage[margin=1.25in]{geometry}
\usepackage{mathrsfs}
\usepackage{mathtools}
\usepackage[shortlabels]{enumitem}
%\usepackage{mathpazo} %A different font
%\usepackage{eulervm} %A different font
\usepackage{hyperref}
\hypersetup{
	colorlinks=true, %set true if you want colored links
	linktoc=all,     %set to all if you want both sections and subsections linked
	linkcolor=red,  %choose some color if you want links to stand out
}

%%---------------------------------------------------------------%%
%%------------------------Custom Commands------------------------%%
%%---------------------------------------------------------------%%
%Some LaTeX commands, like \mathbb{} are used so often that its useful to define a shorter command. Here are some useful examples, but you can add your own. 

\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb R}
\newcommand{\Q}{\mathbb Q}
\newcommand{\C}{\mathbb C}
\renewcommand{\phi}{\varphi}
\newcommand{\defeq}{\overset{\text{def}}{=}}
\newcommand{\ds}{\displaystyle}
\newcommand{\m}{\overline{m}}
\newcommand{\n}{\overline{n}}
\newcommand{\Maps}{\operatorname{Maps}}
\newcommand{\Mag}{\operatorname{Mag}}
\newcommand{\Mat}{\operatorname{Mat}}
\newcommand{\Span}{\operatorname{span}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\nullity}{\operatorname{nullity}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\ch}{\operatorname{ch}}
\newcommand{\ev}{\operatorname{ev}}
\newcommand{\0}{\overline{0}}
\newcommand{\1}{\overline{1}}
\newcommand{\2}{\overline{2}}
\newcommand{\3}{\overline{3}}
\newcommand{\4}{\overline{4}}

%%---------------------------------------------------------------%%
%%---------------------------------------------------------------%%
%%---------------------------------------------------------------%%




%%---------------------------------------------------------------%%
%%---------You can safely ignore anything above this line--------%%
%%---------------------------------------------------------------%%


%%---------------------------------------------------------------%%
%%------------------Document Begins Here-------------------------%%
%%---------------------------------------------------------------%%

\begin{document}
	
	
	%%---------------------------------------------------------------%%
	%%--------------------------Title--------------------------------%%
	%%---------------------------------------------------------------%%
	
	\title{MATH 117: Daily Assignment 6}
	
	\author{WRITE YOUR NAME HERE}
	
	\maketitle
	
	%%---------------------------------------------------------------%%
	%%---------------------------------------------------------------%% 
	%%---------------------------------------------------------------%% 
	
	 See the \href{https://people.ucsc.edu/~jbreland/teaching/SM22_MATH117/daily_assignments.html}{daily assignment webpage} for due dates, templates, and assignment description.
	Try to explain your reasoning and justify your computations for every problem. You should not appeal to any theorems that we have not proved yet. 
	
	%The enumerate enviroment produces numbered lists. The command \item specifies a list item.
	\begin{enumerate}[\bfseries 1.]
		
		%%---------------------------------------------------------------%%
		%%-------------------------Problem 1-----------------------------%%
		%%---------------------------------------------------------------%%
		
		\item Solve Problem 1 from \href{https://people.ucsc.edu/~jbreland/teaching/SM22_MATH117/daily/5/daily5.pdf}{Daily 5} using Theorem 3.2.2 from today's lecture. 
		
		
	
		%%---------------------------------------------------------------%%
		%%--------------------------End----------------------------------%% 
		%%---------------------------------------------------------------%% 	
			
			
		%%---------------------------------------------------------------%%
		%%-------------------------Problem 2-----------------------------%%
		%%---------------------------------------------------------------%%
		
		\item Let $V$ be a vector space. In Friday's lecture, we defined a map $L:V \to V^{**}$ via the rule $L(v) = \ev_v$, where $\ev_v:V^* \to F$ is the function satisfying $\ev_v(f) = f(v)$ for any $f\in V^*$.\footnote{The proof of Theorem 3.1.6 from the lecture actually shows that (assuming the Axiom of Choice, so that every independent set can be extended to a basis in case $V$ has infinite dimension) $L:V\to V^{**}$ is injective, regardless of whether or not $V$ the dimension of $V$ is finite. So $V$ is always isomorphic to a subspace of $V^{**}$. If $V$ is finite-dimensional, then $L$ is an isomorphism because $dim V = \dim V^{**}$.} 
		
		\begin{enumerate}[(a)]
			\item Verify that $\ev_v \in V^{**}$ for all $v\in V$. Thus, $L$ defines a function from $V$ to $V^{**}$.
			
			\item Verify that $L$ is a linear map. 
		\end{enumerate}
		
		
		
		%%---------------------------------------------------------------%%
		%%-------------------------Problem 3-----------------------------%%
		%%---------------------------------------------------------------%%
		
		
		\item Let $V = \R^2$ and let $E = (e_1,e_2)$ denote the standard basis for $V$. In can be shown that $E\otimes E := (e_1 \otimes e_1, e_1 \otimes e_2, e_2\otimes e_1, e_2 \otimes e_2 )$ is a basis for $V\otimes V$. Compute $[(5,3)\otimes (-1,2)]_{E\otimes E}$.\footnote{Hint: Take advantage of the relations in Proposition 3.3.3.}
		
		
		%%---------------------------------------------------------------%%
		%%--------------------------End----------------------------------%% 
		%%---------------------------------------------------------------%% 
		
		
		%%---------------------------------------------------------------%%
		%%-------------------------Problem 4-----------------------------%%
		%%---------------------------------------------------------------%%
		
		
		\item Let $V$ be a vector space. Use the Universal Property of the Tensor Product to show that there is a linear map $L:V\otimes V^* \to F$ satisfying $L(v \otimes f) = f(v)$ for all $v\in V$ and $f \in V^*$. 
		
		
		%%---------------------------------------------------------------%%
		%%--------------------------End----------------------------------%% 
		%%---------------------------------------------------------------%% 
		
		
		

		
		

				
			
		
	\end{enumerate}
	
	%%---------------------------------------------------------------%%
	%%------------------Document Ends Here---------------------------%%
	%%---------------------------------------------------------------%%
	
\end{document}

%%---------------------------------------------------------------%%