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\noindent
\parbox{2in}{\bf Math 5230-101} 
\hfill {\Large \bf Big Quiz \color{red} -- Extended Edition} \hfill
\parbox{2in}{\bf \hfill November 9$^{\mathrm{th}}$\!, 2016}

\force \hfill Hopefully, not too surprising. \hfill \force

\vspace{0.05in}

{\noindent \color{red} A few parts in red were deleted for sake of time.}

\vspace{0.3in}

\noindent {\large\bf Name:} \underline{\hskip 3.0 truein}
\hfill {\bf Be sure to show your work!}

\vspace{0.2in}

\noindent {\bf\large 1.} Let $W = \left\{ \begin{bmatrix} a+b & a+b \\ a-c & a-c \end{bmatrix} \;\Bigg|\; a,b,c \in \mathbb{R} \right\}$. Show $W$ is a subspace using the subspace test. Then again show it is a subspace by recognizing it as a span or a kernel. Find a basis for $W$.

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\noindent {\bf\large 2.} Let $V = \left\{ \begin{bmatrix} a & b  \\ c & d \end{bmatrix} \in \mathbb{R}^{2 \times 2} \;\Bigg|\; a-b=0 \mbox{ and } c-d=0 \right\}$. Show $V$ is a subspace using the subspace test. Then again show it is a subspace by recognizing it as a span or a kernel. Find a basis for $V$.

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\noindent {\color{red}{\bf\large X.} With $W$ and $V$ defined above. Show $W \subseteq V$. Is $W=V$?}

\vspace{1in}

\noindent {\bf\large 3.} Let $U = \left\{ \begin{bmatrix} a & a^2  \\ a^2 & a \end{bmatrix} \;\Bigg|\; a \in \mathbb{R} \right\}$. $U$ is not a subspace. Explain why not.

\vspace{0.25in}


\newpage

\noindent {\bf\large 4.} Let $\alpha = \{ 1,x,x^2 \}$ (note the order) be the standard basis for $P_2 = \{ ax^2+bx+c \;|\; a,b,c \in \mathbb{R} \}$. Let $\beta = \{1,1+x,1+x+x^2\}$.  Show $\beta$ is linearly independent using the definition of linear independence. Find the change of coordinate matrix $[I]_\alpha^\beta$.

\vspace{0.05in}

\noindent {\color{red} Show $\beta$ spans from definition.}

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\noindent {\bf\large 5.} Let $f(x)$ and $g(x)$ be some elements in $P_2$. How do I know for certain that $\delta = \{ f(x),g(x) \}$ is not a basis for $P_2$? What can be said? What's possible?

\vspace{1.5in}

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\noindent {\bf\large 6.}  Let $T:P_2 \to \mathbb{R}^{2 \times 2}$ where $T(ax^2+bx+c) = \begin{bmatrix} a & a+b \\ a-b & 0 \end{bmatrix}$. Prove that $T$ is linear. Find a standard coordinate matrix for $T$. Find a basis for $T$'s kernel and range. State $T$'s rank, nullity and whether it is one-to-one, onto, both or neither.

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\noindent {\bf\large 7.}  Suppose $T:V \to W$ is a linear map. Let $\alpha$ and $\gamma$ be bases for $V$, and $\beta$ and $\kappa$ be bases for $W$. Also, suppose that\dots
$$A=[T]_\alpha^\beta = \begin{bmatrix} 1 & 2 & 3 \\ 1 & -1 & 0 \end{bmatrix} \qquad B=[I]_\gamma^\alpha = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{bmatrix} \qquad C=[I]_\kappa^\beta = \begin{bmatrix} 1 & 2 \\ 0 & -1 \end{bmatrix}$$

\vspace{0.2in} 

$\mathrm{dim}(V)=$ \underline{\hspace*{1in}} \qquad $\mathrm{dim}(W)=$ \underline{\hspace*{1in}}.

\vspace{0.2in}

Find $[T]_\gamma^\kappa$.

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\newpage

\noindent {\bf\large 8.}  Give a quick argument using bases to prove that $\dim(W) \leq \dim(V)$ if $W$ is a subspace of $V$.

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\noindent {\bf\large 9.} Let $U$ and $W$ be subspaces of some vector space $V$. What does $V = U \oplus W$ mean? (Give a definition.)

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\noindent {\bf\large 10.} Let $\alpha =  \{ \B u \}$ be a basis for $U$. Let $\beta = \{ \B v, \B w \}$ be a basis for $W$. Assume that $V=U \oplus W$. Prove that $\gamma = \alpha \cup \beta$ is a basis for $V$. [Do not just say, ``It's a theorem.'']

\vfill

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