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\begin{document}
\title{Problem Set \#1}
\author{Due September 8}
\maketitle

1. Find 2 distinct bases for $\mathbb Q^2$.

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2. Find all 3 bases of $\mathbb F_2^2$.

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3. Are the vectors $(1,1,1)$, $(0,-1,2)$, $(2,1,3)$, and $(4,0,11)$ linearly independent? If not, find a dependence relation.

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4. Let $a,b\in\mathbb F$ and let $\vec{v},\vec{w}\in V$. Show that 
\[
(a+b)(\vec{v}+\vec{w})=a\vec{v}+a\vec{w}+b\vec{v}+b\vec{w}.
\]

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5. Let $C(\mathbb R)$ be the collection of all continuous functions from $\mathbb R$ to itself. Let $C_+$ denote the subset of ``even'' functions:
\[
C_+=\{f | f(-x)=f(x)\},
\]
and let $C_-$ denote the ``odd'' functions:
\[
C_-=\{f | f(-x)=-f(x)\}.
\]
Show that both are subspaces of $C(\mathbb R)$. Show also that $C(\mathbb R)=C_+\oplus C_-$.

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6. Let $V$ be the vector space of all infinite sequences of real numbers. Show that the following are subspaces:
\begin{enumerate}
\item $\ell_1$, the subset of absolutely convergent sequences: \[\ell_1=\{(a_1,\dots) | \sum |a_n|<\infty\}.\]
\item $z_0$, the subset of eventually zero sequences.
\item $c_0$, the subset of sequences which converge to zero.
\item $c$, the subset of sequence which converge.
\item $\ell_\infty$, the subset of bounded sequences: \[\ell_\infty=\{(a_1,\dots) | \sup |a_n|<\infty\}.\]
\end{enumerate}
\end{document}