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\noindent
\parbox{2.25in}{\bf Math 5230}
\hfill {\Large \bf Homework \#6} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Oct. $14^\mathrm{th}$, 2022}

\vspace{0.2in}

\begin{description}
\item[\#1 Bases] Consider $W = \left\{ \begin{bmatrix} -3a-4b+c & b-c & -4a-6b+2c \\ a+b & 2b-2c & a+b \end{bmatrix} \;\Big|\; a,b,c \in \mathbb{R} \right\}$ and 

$V = \left\{ \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix} \;\Bigg|\; \begin{array}{ccc} a+3b+2c+4d+e+7f & = & 0 \\ 2a+6b+c+5d-e+5f & = & 0 \\ 3a+9b+c+7d-2e+6f & = & 0 \end{array} \right\}.$
    \begin{enumerate}[(a)]
    \item Both $W$ and $V$ are subspaces of $\mathbb{R}^{2 \times 3}$. Prove that they are subspaces by re-expressing each as either a span or kernel (i.e., null space). Pick the most convenient description.
    
    {\it Note:} Re-expressing as a span or kernel would prove we have a subspace since spans and kernels are always subspaces. 
    \item Prove that $W \subseteq V$.
    \item Find a basis for $V$.
    \item Find a basis for $W$ and then extend this to a basis for all of $V$.
    \end{enumerate}
    
\item[\#2 Linear Transformation Basics] Define $T:\mathbb{R}^{2 \times 2} \to \mathbb{R}[x]$ as follows:
     $$T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = (a+2b+3d)x^2+(-a-2b+c+d)x+(a+2b+c+7d).$$
     Note that $T$ is a linear transformation (you don't need to prove this).
     \begin{enumerate}[(a)]
     \item Write down the standard coordinate matrix for $T$ and find its RREF. \\ {}[You may truncate the infinitely many rows of zeros.]
     \item Find a basis for the kernel and range of $T$. 
     \item What is the nullity and rank of $T$? Is $T$ 1-1, onto, both, neither? 
     \end{enumerate}
     
\item[\#3 Trace] Let $A \in \mathbb{F}^{n \times n}$ ($\mathbb{F}$ is a field) and let $a_{ij}$ be the $(i,j)$-entry of $A$. Recall that $\mathrm{tr}(A)=\sum\limits_{i=1}^n a_{ii}$ (the {\it trace} of $A$ is the sum of its diagonal entries). 
   \begin{enumerate}[(a)]
   \item Prove that $\mathrm{tr}:\mathbb{F}^{n \times n} \to \mathbb{F}$ is linear.
   \item Identify the kernel, range, nullity, and rank of the trace map.
   \end{enumerate}     

\item[\#4 Linear] Let $\mathcal{I},\mathcal{D}:\mathbb{R}[x] \to \mathbb{R}[x]$ be defined by $\mathcal{I}(f(x)) = \int_0^x f(t)\,dt$ and $\mathcal{D}(f(x))=f'(x)$. 
    \begin{enumerate}[(a)]
    \item Show $\mathcal{I}$ is linear. Identify its kernel and range. Is $\mathcal{I}$ one-to-one? onto? an isomorphism?
    \item Show $\mathcal{D}$ is linear. Identify its kernel and range. Is $\mathcal{D}$ one-to-one? onto? an isomorphism?
    \item Describe $\mathcal{I} \circ \mathcal{D}$ and $\mathcal{D} \circ \mathcal{I}$. 
    \end{enumerate}
\end{description}

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