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\begin{document}

\begin{center}
\textbf{Math 3013}\\[0pt]
SECOND EXAM \\[0pt]
November 3, 2020
\end{center}

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

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

(a) (10 pts) Determine if these vectors are linearly independent.

\vspace{3in}

(b) (5 pts) What is the dimension of the subspace generated by these
vectors? \vspace{.5in}

2. Write the definitions (as stated in class) of the following notions. (5
pts each)

(a) A \textbf{subspace} $W$ of $\mathbb{R}^{n}.$

\vspace{1in}

(b) A \textbf{basis }for a subspace  $W$ of $\mathbb{R}^{n}$.

\vspace{1in}

(c) A set of \textbf{linearly independent vectors }

\vspace{1in}

(d) A \textbf{linear transformation} 

\newpage

3. Given that the matrix $\displaystyle\mathbf{A}=\left[ 
\begin{array}{cccc}
2 & 4 & 0 & 2 \\ 
-1 & -2 & 1 & -3 \\ 
0 & 0 & 1 & 2  
\end{array}%
\right] $ \quad  row reduces to \quad $\left[ 
\begin{array}{cccc}
1 & 2 & 0 & 1 \\ 
0 & 0 & 1 & -2 \\ 
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 $\mathbf{A}$?

\vspace{.5in}

4. Consider the following linear transformation: 

\hspace{.5in} $T:\mathbb{R}^{3}
\rightarrow \mathbb{R}^{3}:$ $T\left( [x_{1}\; ,\; x_{2}\; ,\; x_{3} \right) =\left[
x_{1} +2 x_{2}+x_{3}\; , \; x_{1} + x_{2} \; , \;x_{1}-x_{3}\right] $.

(a) (10 pts) Find a matrix that represents $T$.

\vspace{1.5in}

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

\vspace{1.4in}

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


\newpage

5. (15 pts) Use cofactor expansions to compute the determinant of $\mathbf{A} = 
\left[ 
\begin{array}{cccc}
1 & 0 & 0 & 1\\ 
1 & 0 & 1 & 1 \\ 
1 & 2 & 1 & 1 \\
1 & 0 & 0 & 2
\end{array}%
\right] $

\vspace{4in}



6. (15 pts) Use the row reduction method to determine the determinant of $\mathbf{A} = 
\left[
\begin{array}{cccc}
0 & 0 & 0 & 1 \\
0 & 2 & 2 & 2 \\
0 & 2 & 1 & -1 \\
1 & 1 & 1 & 1
\end{array}
\right]$


\end{document}
