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	\title{MATH 117: Daily Assignment 2}
	
	\author{Carl Friedrich Gauss}
	
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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/daily_assignments.html}{daily assignment webpage} for due dates, templates, and assignment description.
	
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	\begin{enumerate}[\bfseries 1.]
		
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		\item Let $F$ be any field. A \emph{Fibonacci sequence} in $F$ is a function $f:\N \to F$ defined recursively by letting $f(0),f(1)$ be elements of $F$ and then setting $f(n+2) = f(n+1) + f(n)$ for all $n\in \N$. Show that the set $\mathcal{F}$ of all Fibonacci sequences in $F$ is a subspace of $F^{\N}$. Then compute the dimension of this space.
		
		\begin{proof}[Solution] Your solution can go here.
		\end{proof}
	
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		\item For each part, determine whether the set of vectors $S$ is a spanning set for the vector space $V$ over the field $F$. If $S$ is a spanning set, determine whether or not it is a basis\footnote{Hint: use a Proposition from Section 1.4! }. Justify your answers. 
		
		\begin{enumerate}[(a)]
			
			\item $F = \Q$, $V = \Q(\sqrt{2}) = \{a + b\sqrt{2} \in \R : a,b \in \Q\}$, $S = \{1-\sqrt{2}, 4\}$. 
			\item $F = \Z_2$, $V = \left\{\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix} \in F^{2\times 2} : a_{11} + a_{22} = 0  \right\}$, $S = \left\{\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix},\begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix}, \begin{pmatrix}1 & 0\\ 1 & 1\end{pmatrix}\right\}$
			
			\item $F = \Z_2$, $V = F_{2}[x] = \{a_0 + a_1x + a_2x^2: a_0,a_1,a_2 \in F\}$, $S = \{1+x, 1+x^2\}$
		\end{enumerate}
	
		\begin{proof}[Solution] Your solution can go here.
		\end{proof}
	
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		\item Let $M:=\Mag_3(\R)$ denote the set of $3 \times 3$ magic squares with entries from $\R$.
		
		\begin{enumerate}[(a)]
			\item Show that $M$ is a subspace of $\R^{3\times 3}$.
			
			\item Find a basis for $M$.\footnote{You will need to solve a large system of equations - you may use a computer algebra system (CAS) to do this part of the computations. }
			
			\item What is the dimension of $M$?
		\end{enumerate}		
		
		\begin{proof}[Solution] Your solution can go here.
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		\item (optional) Let $M:=\Mag_3(F)$ denote the set of $3 \times 3$ magic squares with entries from a field $F$.
		
		\begin{enumerate}[(a)]
			\item Convince yourself that the set $M_0$ of magic square with magic sum 0 is a subspace of $M$.
			
			\item Find a basis for $M_0$ when $F = \Z_p$, the field with $p$-elements. Does your answer depend on $p$?
			
			\item What is the dimension of $M_0$?
		\end{enumerate}		
		
		\begin{proof}[Solution] Your solution can go here.
		\end{proof}
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