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\begin{center}
\textbf{Math 3013.21403} \\
SECOND EXAM \\ 
March 28, 2022
\end{center}

\textbf{Calculators and/or notes are not permitted for this exam.}

\begin{flushright}
Name:\underline{\hspace{2.5 in}}
\end{flushright}


1. Define, precisely, the following notions (where $V$, $W$ are to be regarded
as general vector spaces). space $V$).

(a) (5 pts) a \textbf{subspace} of $V$ is ...

\vspace{1in}

(b) (5 pts) a \textbf{basis} for a vector space $V$ is ...

\vspace{1in}

(c) (5 pts) a \textbf{set of linearly independent vectors} in $V$ is ...

\vspace{1in}

(d) (5 pts) a \textbf{linear transformation} from a vector space $V$ to vector
space $W$ is ...


\vspace{1in}

2. (10 pts) Prove or disprove that the points on the set $S=\left\{
\left[  x,y\right]  \in\mathbb{R}^{2} \mid y = x+1 \right\}  $ is a
subspace of $\mathbb{R}^{2}$.

\newpage


3. (10 pts) Let $W=span\left([1,1,1,1],[1,-1,1,0],[2,0,2,1]\right) \subset \mathbb{R}^4$. Find a basis for $W$

\vspace{2in}

4. Consider the following matrix: \quad$\displaystyle
\mathbf{A}=\left[
\begin{array}
[c]{cccc}%
2 & 0 & 0 & 2\\
2 & 2 & 0 & 4\\
0 & 2 & 0 & 2
\end{array}
\right]  $


(a) (10 pts) Row reduce this matrix to reduced row echelon form

\vspace{3in}

(b) (5 pts) Find a basis for the row space of $\mathbf{A}$.

\vspace{.5in}

(c) (5 pts) Find a basis for the column space of $\mathbf{A}$.

\vspace{0.75in}

(d) (5 pts)\ Find a basis for the null space of $\mathbf{A}$.

\vspace{1in}

\bigskip(e) (5 pts) What is the rank of $\mathbf{A}$?

\vspace{.5in}



\newpage

5. (10 pts)  Let $\mathbf{A}$ be an $n\times m$ matrix. Show that the function 
$T_{\textbf{A}} : \mathbb{R}^{m} \rightarrow \mathbb{R}^{n} : 
T\left( \mathbf{x} \right) = \mathbf{Ax}$ (via matrix multiplication on the right) 
is a linear transformation. 


\vspace{2.5in}

6. Consider the linear transformation $T:\mathbb{R}^{3}\rightarrow
\mathbb{R}^{2}:T\left(  \left[x_{1},x_{2},x_{3}\right]\right)  =\left[ x_{1}-x_{3}, x_{2}+x_{3} \right]$.

(a) (10 pts) Find the matrix $\mathbf{A}_{T}$ corresponding to $T$:

\vspace{2in}

(b) (5 pts)\ Find a basis for $range(T)\equiv\left\{  \mathbf{y}\in
\mathbb{R}^{2}\mid\mathbf{y}=T(\mathbf{x})\text{ for some }\mathbf{x}%
\in\mathbb{R}^{3}\right\}  $.

\vspace{1in}

(c) (5 pts) Find a basis for $ker(T)\equiv\left\{  \mathbf{x}\in\mathbb{R}%
^{4}\mid T(\mathbf{x})=\mathbf{0}\right\}  $

\vspace{1in}




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