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\title{Linear Algebra - \LaTeX\ HW 1}
\author{Mark McClure}
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\noindent This 20 point homework is due Monday, October 21. I expect you to type your soluiton in \LaTeX\, print it out and turn in the hard copy.
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\item[Problem 1:]\text{ }\\
Let $V$ be a vector space and let $\vec{u}, \vec{v}, \vec{w} \in V$. Use the axioms and basic properties of vector spaces to show that if
$$\vec{u}+\vec{w} = \vec{w} + \vec{v},$$
then $\vec{u} = \vec{v}$. \\
\textit{Comment}: After restating the assumptions, try to structure your proof as a string equalities that are justified by specific statements, as the text does on pages 203-205 in its proofs of the vector space properties.
\item[Problem 2:]\text{ }\\
Let $V$ denote the set of all polynomials whose degree is at most $n$. Find a basis for $V$ and \textit{prove} that your basis is, in fact, a basis. \\
\textit{Comment}: I think there's a natural choice for the basis. Given that natural choice, it's pretty easy to show that the set spans $V$. It's a bit trickier to show that the set is linearly independent. You may assume the fact that a polynomial of degree $n$ has at most $n$ roots.
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