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\homework{16: Linear Algebra and Field Extensions}{Due Friday, Week 9}


Solve \textbf{one} of the following \textbf{three} problems.    As always, prove your claims/have fun! 


\begin{enumerate}
\item In class, we claimed that if $V$ is a vector space and $A$ is a subset of $V$, then the \textbf{span} of $A$, $span(A)$, is a vector space.  Prove this claim!
\item In class, we claimed that the collection of all polynomials with real-valued coefficients $\mathbb{R}[x]$ was a vector space.  We verbally talked about why this was true, but didn't do a formal proof.

Write down a formal proof of this claim (i.e.\ check that this object forms a vector space over $\mathbb{R}$!)
\item In class, we made the following claim:
\begin{thm}
Suppose that $V$ is a vector space with two bases $B_1 = \{\vec{v_1}, \ldots \vec{v_n}\}$, $B_2 = \{\vec{w_1}, \ldots \vec{w_m}\} $ both containing finitely many elements..  Then these sets have the same size: i.e.\ $|B_1| = |B_2|.$
\end{thm}
Prove this claim!
\end{enumerate}


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