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\noindent
\parbox{2.25in}{\bf Math 5230}
\hfill {\Large \bf Homework \#5} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Oct. $10^\mathrm{th}$, 2016}

\vspace{0.2in}

\noindent
\underline{Notation:} $V$ is a vector space over a field $\mathbb{F}$. Do not assume $V$ is finite dimensional (unless told otherwise). Let $W_1$ and $W_2$ be subspaces of $V$. 

\vspace{0.1in}

\begin{description}
\item[\#1 Vector Space Definition] Using the definition -- and that's all.

\begin{enumerate}[(a)]
\item In horrifying detail (go through all of the axioms), show that $\mathbb{R}^2 = \{ (a,b) \;|\; a,b \in \mathbb{R} \}$ is a vector space over $\mathbb{R}$.
\item Let $\B u, \B v, \B w \in V$. Prove that $\B u+\B v = \B u+\B w$ implies that $\B v = \B w$. Do this {\it only using axioms}. Cite the axioms being used. 
\end{enumerate}

\item[\#2 Easy Subspace] Show that $W = \{ (x,y) \in \mathbb{R}^2 \;|\; x+2y=0 \}$ is a subspace of $\mathbb{R}^2$. Then explain why $B = \{ (x,y) \in \mathbb{R}^2 \;|\; x+2y=3 \}$ is not a subspace.

\item[\#3 Subspace] Recall that a function $f:\mathbb{R} \to \mathbb{R}$ is called {\bf even} if $f(-x)=f(x)$. A function is called {\bf odd} if $f(-x)=-f(x)$. Recall that $V=\mathbb{R}^\mathbb{R}$ is the vector space of all functions from the reals to the reals.

\begin{enumerate}[(a)]
   \item Let $E$ be the set of all even functions in $\mathbb{R}^\mathbb{R}$. Show that $E$ is a subspace.
   \item Let $O$ be the set of all odd functions in $\mathbb{R}^\mathbb{R}$. Show that $O$ is a subspace.
   \item Notice that for any $f \in \mathbb{R}^\mathbb{R}$, $g(x) = \dfrac{f(x)+f(-x)}{2}$ is even and $h(x) = \dfrac{f(x)-f(-x)}{2}$ is odd. Prove that $\mathbb{R}^\mathbb{R} = E \oplus O$. Demonstrate how $f(x) = x^3-5x^2+\cos(x)+4\sin(x)-12$ decomposes.
\end{enumerate}

{\it ``Fun'' Side Note:} When we decompose a function into an even function plus an odd function, we call these parts it's even and odd parts. The even and odd parts of the exponential function, $e^x$, are called $\cosh(x)$ and $\sinh(x)$ (the hyperbolic cosine and sine functions).

\item[\#4 Unions Don't Work] Show that $W_1 \cup W_2$ is a subspace if and only if $W_1 \subseteq W_2$ or $W_2 \subseteq W_1$.

\item[\#5 Linear Independence and Spanning] Let $V$ be a vector space over $\mathbb{F}=\mathbb{Q}$ (the rational numbers) with basis $\beta = \{\B u, \B v, \B w\}$ (distinct vectors).

\begin{enumerate}[(a)]
    \item Prove that $S = \{\B u+\B v, \B v +\B w, \B w +\B u \}$ is linearly independent (using the definition of independence).
    \item Prove that $T = \{\B u+\B v, \B u - \B v, \B w \}$ spans $V$ (using the definition of span).
    \item Consider $S \cup T$ and $S \cap T$. Decide whether each set is linearly independent, spans $V$, both spans and is linearly independent, or neither. Give proof of each assertion.
\end{enumerate}

\item[\#6 Dimension] For convenience, let's assume $V$ is finite dimensional. Note that $W_1+W_2 = \{ u+v \;|\; u \in W_1 \mbox{ and } v \in W_2 \}$. Also, recall that $V=W_1 \oplus W_2$ if $V=W_1 + W_2$ and $W_1 \cap W_2 = \{\B 0\}$.

\begin{enumerate}[(a)]
   \item Prove that $\dim(W_1 \cap W_2) \leq \min \{ \dim(W_1), \dim(W_2) \}$.
   \item Prove that $\dim(W_1 + W_2) \geq  \max \{ \dim(W_1), \dim(W_2) \}$.
   \item If $V=W_1 \oplus W_2$, what is $\dim(V)$? Justify your answer.
\end{enumerate}

As a general bit of advice for problems like this, assume (without loss of generality) that $\dim(W_1) = m \leq \dim(W_2)=n$ so that (for example) the minimum of the two dimensions is $m$ and the maximum is $n$.
\end{description}

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