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	\title{MATH 117: Daily Assignment 1}
	
	\author{Carl Friedrich Gauss}
	
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	Some hints/comments for this assignment may be 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.
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		\item Sign up for the Zulip discussion forum. Check your @ucsc.edu email for an invite to join or use the invite link on the canvas page. Then complete the following tasks:
		
		\begin{enumerate}[(a)]
			\item Introduce yourself to the class on Zulip. Make a post on the \href{https://math117.zulipchat.com/#narrow/stream/398694-introductions}{introductions stream} using your first and last name as the title. Respond to at least two other posts on this stream
			
			\item Create a \LaTeX\ example for your classmates. It can be as simple as mine (how to: create a matrix), but the topic should be unique. Post your example on the \href{https://math117.zulipchat.com/#narrow/stream/398695-LaTeX}{\LaTeX\ stream}. Make sure to use the Zulip latex code block to display your raw code (click view source on my post to see the syntax). Title your post as follows: ``how to: (your topic here)".  
		\end{enumerate} 
		
		
		\item Consider the set $\Z_7 = \{\0, \1, \2, \3, \4, \5, \6\}$ of residue classes of integers modulo 7.
		
		\begin{enumerate}[(a)]
			\item Construct the multiplication table for the group $(\Z_7\setminus \{\0\},\cdot)$ where $\cdot$ is defined using representatives: $\m \cdot \n := \overline{mn}$. 
			
			\item Use part (a) to find the multiplicative inverse of every nonzero element of $\Z_7$. 
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		\item Let $V$ be a vector space over a field $F$. Using only the definitions, prove Proposition 1.2.2: for all $v\in V$ and $a \in F$,
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			\item $0v = 0$;
			\item $(-a)v = -(av)$;
			\item $a0 = 0$; and
			\item $av = 0$ implies $a = 0$ or $v = 0$. 
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		\item Let $C(\R)$ be the real vector space\footnote{$C(\R)$ is a subspace of the real vector space $\R^\R = \Maps(\R,\R)$} of all continuous functions $f:\R \to \R$. Determine which of the following are subspaces of $C(\R)$. Make sure to justify your reasoning.
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			\item $\{f : f \text{ is twice differentiable and } f''(x) - 2f'(x) + 3f(x) =0 \text{ for all }x \in \R\}$.
			\item $\{g : g \text{ is twice differentiable and } g''(x) = g(x) + 1 \text{ for all }x \in \R\}$.
			\item $\{h : h \text{ is twice differentiable and } h''(0) = 2h(1)\}$.
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