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	\title{Weekly Assignment 1}
	
	\author{Carl Friedrich Gauss\\
		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.
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		\item Let $F$ be a field and let $V=F$. Denote the additive and multiplicative identities of $F$ by $0_F$ and $1_F$, respectively. For $u,v\in V$ and $\alpha \in F$, define vector addition by $u \oplus v : = u+v - 1_F$ and scalar multiplication by $\alpha \odot u := \alpha u - \alpha + 1_F$. Prove that $(V, \oplus,\odot)$ is an $F$-vector space.\footnote{You need to specify a zero vector $0_V$ and the additive inverse $\ominus u$ of $u \in F$, and then verify the several defining conditions of a vector space.} 
		
		\begin{proof} Write your proof here.
			
		\end{proof}
		
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		\item Suppose that $W_1,\ldots,W_n$ are subspaces of a vector space $V$ over a field $F$. Prove that
		\[
		\sum_{i=1}^n W_i = \left\{\sum_{i=1}^nw_i : w_i \in W_i\text{ for all } i=1,\ldots,n\right\}.
		\]
		
		\begin{proof} Write your proof here.
			
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		\item Prove Proposition 1.4.8: a subset $B$ of a vector space $V$ is a basis if and only if $B$ is a minimal\footnote{A minimal spanning set is a spanning set that does not properly contain any other spanning set.} spanning set.
		
		\begin{proof} Write your proof here.
			
		\end{proof}
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		\item Let $M$ and $N$ be finite-dimensional subspaces of a (not necessarily finite dimensional) vector space $V$. Prove the following equation:
		\[
		\dim(M) + \dim(N) = \dim(M+N) + \dim(M\cap N).
		\]
		
		\begin{proof} Write your proof here.
			
		\end{proof}
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