%%---------------------------------------------------------------%%
%%-----------------------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}}
\renewcommand{\span}{\operatorname{span}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\nullity}{\operatorname{nullity}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\ch}{\operatorname{ch}}

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




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


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

\begin{document}
	
	
	%%---------------------------------------------------------------%%
	%%--------------------------Title--------------------------------%%
	%%---------------------------------------------------------------%%
	
	\title{MATH 117: Daily Assignment 3}
	
	\author{Carl Friedrich Gauss}
	
	\maketitle
	
	%%---------------------------------------------------------------%%
	%%---------------------------------------------------------------%% 
	%%---------------------------------------------------------------%% 
	
	Some hints for this assignment are written in the footnotes. See the \href{https://people.ucsc.edu/~jbreland/teaching/SM23_MATH117/daily_assignments.html}{daily assignment webpage} for due dates, templates, and assignment description.
	
	%The enumerate enviroment produces numbered lists. The command \item specifies a list item.
	\begin{enumerate}[\bfseries 1.]
		
		%%---------------------------------------------------------------%%
		%%-------------------------Problem 1-----------------------------%%
		%%---------------------------------------------------------------%%
		
		\item Compute $\rank(L)$ and $\nullity(L)$ for each of the following linear maps. You must justify your reasoning.
		
		\begin{enumerate}[(a)]
			
			\item $L:F_2[x]\to F_2[x]$, $L(a + bx + cx^2) = b + 2cx$ where $F$ is any field.\footnote{Hint: the answer depends on the characteristic of the field! Handle the case $\ch F=2$ separately. }
			
			\item $L:F^{2\times 2} \to F$, $L(A) = \tr(A)$ where $F$ is any field. Here, $\tr(A)$ denotes the \emph{trace} of the matrix $A$, i.e., the sum of the element on the main diagonal. 
			
%			\item $L:\Q(\sqrt{2}) \to \Q$, $L(a+b\sqrt{2}) = a+b$. Here, we view $\Q(\sqrt{2})$ and $\Q$ as vector spaces over $\Q$. 
%			
%			\item $L:F^3 \to F^3$, $L(a,b,c) = (0,b,c)$ where $F$ is any field. 
		\end{enumerate}
		
		\begin{proof}[Solution] Your solution can go here.
		\end{proof}
	
		%%---------------------------------------------------------------%%
		%%--------------------------End----------------------------------%% 
		%%---------------------------------------------------------------%% 	
			
		
		
		
		%%---------------------------------------------------------------%%
		%%-------------------------Problem 2-----------------------------%%
		%%---------------------------------------------------------------%%
		
		\item For each part, you are given a finite-dimensional $F$-vector space $V$ with ordered basis $B$ and a vector $v \in V$. Compute $[v]_B$. 
		
		\begin{enumerate}[(a)]
			\item $V = \R^3$, $F = \R$, $B = ((1,1,2), (2,3,2), (1,0,1) )$, $v = (-2,1,4)$.
			
			\item $V = F_2[x]$, $F$ is any field with $\ch F \neq 2$\footnote{This condition is required for the vectors in $B$ to be independent.}, $B = (1+x, 1+x^2,x+x^2)$, $v=v(x) = a + bx + cx^2$ where $a,b,c \in F$.
			
			\item $V = \C$, $F = \R$, $B = (i,1-i)$, $v = x+iy$. Here $i$ denotes the imaginary unit in $\C$. 
			
%			\item $V = \Mag_3(\R)$ (see Daily 1.3), $F = \R$, 
%			$B = \left\{
%			\begin{psmallmatrix}
%				0 & 1 & -1\\ 
%				-1 & 0 & 1\\ 
%				1 & -1 & 0
%			\end{psmallmatrix},
%			\begin{psmallmatrix}
%				-1 & 1 & 0\\
%				1 & 0 & -1\\
%				0 & -1 & 1
%			\end{psmallmatrix},
%			\begin{psmallmatrix}
%				1 & 1 & 1\\
%				1 & 1 & 1\\
%				1 & 1 & 1
%			\end{psmallmatrix}\right\}$, 
%			$v = 
%			\begin{psmallmatrix}
%				2 & 7 & 6\\
%				9 & 5 & 1\\
%				4 & 3 & 8
%			\end{psmallmatrix}$.
		\end{enumerate}		
		
		\begin{proof}[Solution] Your solution can go here.
		\end{proof}
		%%---------------------------------------------------------------%%
		%%--------------------------End----------------------------------%% 
		%%---------------------------------------------------------------%%
		
		
		
		%%---------------------------------------------------------------%%
		%%-------------------------Problem 3-----------------------------%%
		%%---------------------------------------------------------------%%

		\item 
		\begin{enumerate}[(a)]
			\item Construct a linear map $L:\R^3 \to \R^3$ which maps the plane $\{(x,y,z)\in \R^3 : z = 0\}$ bijectively onto the plane $\{(x,y,z)\in \R^3 :  3x + 2y + z = 0\}$.\footnote{Theorem 2.3.1 my be useful here. }
			
			\item Construct a linear map $L:\R^3 \to \R^3$ which maps the plane $\{(x,y,z)\in \R^3 : x-y-z = 0\}$ bijectively onto the plane $\{(x,y,z)\in \R^3 :  x-3z = 0\}$.
		\end{enumerate}
	
		\begin{proof}[Solution] Your solution can go here.
		\end{proof}
	
		%%---------------------------------------------------------------%%
		%%--------------------------End----------------------------------%% 
		%%---------------------------------------------------------------%% 

		%%---------------------------------------------------------------%%
		%%-------------------------Problem -----------------------------%%
		%%---------------------------------------------------------------%%
		
		\item (optional) Try to construct a choice function on $\mathcal{P}(\R)\setminus \{\emptyset\}$, the set of all nonempty subsets of the real numbers. Then give an informal argument (not a proof) for why it is not possible. 
		
		\begin{proof}[Solution] Your solution can go here.
		\end{proof}
		
		%%---------------------------------------------------------------%%
		%%--------------------------End----------------------------------%% 
		%%---------------------------------------------------------------%% 		
		 

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

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