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\begin{document}

\begin{center}
\textbf{Math 3013.62663}\\[0pt]
SECOND EXAM \\[0pt]
October 22, 2021
\end{center}

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


1. Complete the following mathematical definitions

(a) (5 pts) A \textbf{subspace} of a vector space $V$ is $\ldots$ 

\vspace{2in}

(b) (5 pts) A \textbf{basis} for a subspace $W$ is $\ldots$


\vspace{2in}


(c) (5 pts) A set of vectors $\left\{ \mathbf{v}_{1}, \ldots , \mathbf{v}_{k} \right\}$ 
is \textbf{linearly independent} if $\ldots$


\vspace{2in}

(d) (5 pts) A function $T:\mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ is a \textbf{linear
transformation} if $\ldots$


\newpage


2. Consider the vectors 
$\left\{[0,1,-1,2],[1,1,1,1],[2,1,3,0] \right\} \in \mathbb{R}^{4}$

(a) (5 pts) Determine if these vectors are linearly independent.

\vspace{2in}

(b) (5 pts) What is the dimension of the subspace generated by these
vectors?

\vspace{.5in}




3. Given that the following matrix: \quad $\displaystyle\mathbf{A}=\left[ 
\begin{array}{cccc}
1 & 0 & 1 & 3 \\ 
2 & 0 & 2 & 5 \\ 
1 & 0 & 1 & 3 \\ 
\end{array}%
\right] $ \quad  row reduces to \quad $\left[ 
\begin{array}{cccc}
1 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 1 \\ 
0 & 0 & 0 & 0 \\ 
\end{array}%
\right] $

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

\vspace{.8in}

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

\vspace{.8in}

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

\vspace{1.2in}

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

\newpage

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

(a) (5 pts) Find the matrix $\mathbf{A}_{T}$ such that 
$\mathbf{A}_{T} \mathbf{x} = T\left(\mathbf{x}\right)$.

\vspace{1.5in}

(b) (5 pts) Find a basis for the range of $T$.

\vspace{1in}

(c) (5 pts) Find a basis for the kernel of $T$.


\vspace{1.5in}

5. Consider the matrix $\mathbf{A} = \left[ 
\begin{array}{cccc}
0 & 0 & 0 & 3 \\ 
0 & 1 & -3 & 0 \\ 
1 & 0 & 0 & 0 \\ 
0 & 0 & -2 & 0%
\end{array}%
\right] $

(a) 7 pts) Compute $\det \left( \mathbf{A} \right)$ via  a cofactor expansion
along the third row.

\vspace{1.5in}

(b) (8 pts) Compute $\det \left( \mathbf{A} \right)$ by row reducing $\mathbf{A}$ to
an upper triangular matrix.

\newpage


6. (10 pts) Use Cramer's Rule to solve
\begin{eqnarray*}
 x_{1} + 2x_{2} &=& 3 \\
-x_{1} + x_{2} &=& 3
\end{eqnarray*}

\vspace{3in}

7. (10 pts) Find all the cofactors of $\mathbf{A} = \left[
\begin{array}{cc}
2 & 2 \\
3 & 4  
\end{array} \right]$
and then use these cofactors to compute $\mathbf{A}^{-1}$

\end{document}
