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\begin{center}
Math 3013.69033\\[0pt]
Final Exam\\[0pt]
6:00pm - 8:00pm CDT, December 10, 2020\\[0pt]

\end{center}

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


1. Give the definitions of the following linear algebraic concepts:

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

\vspace{1in}



(b) (5 pts) a set of \textbf{linearly independent} vectors

\vspace{1in}

(c) (5 pts) a \textbf{linear transformation} between two vector spaces $V$ and
$W$.

\vspace{1in}


(d) (5 pts) a \textbf{basis} for a subspace $W$ of a vector space $V$

\vspace{1in}


2. For each of the following augmented matrices, describe the solution space
of the corresponding linear system. (Determine if there are solutions; and, if
there are solutions, how many free parameters are needed to describe the
general solution.)

(a) (5 pts) $\left[
\begin{array}
[c]{cccc}%
1 & 0 & 2 & 2\\
0 & 1 & 0 & 1\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{array}
\left\vert
\begin{array}
[c]{c}%
1\\
2\\
0\\
0
\end{array}
\right.  \right]  $

(b) (5 pts)$\left[
\begin{array}
[c]{ccc}%
1 & 0 & 4 \\
0 & 2 & 1 \\
0 & 0 & 1 \\
0 & 0 & 0 
\end{array}
\left\vert
\begin{array}
[c]{c}%
1\\
2\\
1\\
0
\end{array}
\right.  \right]  $

(c) (5 pts) $\left[
\begin{array}
[c]{ccccc}%
0 & -1 & 0 & 1 & 2\\
0 & 0 & 2 & 0 & 1\\
0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0
\end{array}
\left\vert
\begin{array}
[c]{c}%
1\\
0\\
1\\
0
\end{array}
\right.  \right]  $

\newpage

3. (10 pts)\ Compute the inverse of $\mathbf{A}=\left[
\begin{array}
[c]{ccc}%
1 & 2 & 0\\
1 & 3 & 0\\
1 & 2 & 1
\end{array}
\right]  $

\vspace{3in}

4. (10 pts) Let $W=\left\{  \left[  x,y\right]  \in\mathbb{R}^{2}\mid
x+2y=3\in\mathbb{R}\right\}  $. Prove or disprove that $W$ is a subspace of
$\mathbb{R}^{2}$.

\vspace{2in}

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

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

(b) (10 pts) Find a basis for the kernel of $T$ (i.e. the set of vectors
$\mathbf{x}$ such that $T\left(  \mathbf{x}\right)  =\mathbf{0}$).

\newpage

6. Consider the vectors $\left\{  [1,-2,2,1],[1,-1,3,1],[0,1,1,0],\left[
2,-3,5,2\right]  \right\}  \in\mathbb{R}^{4}$

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

\vspace{2.5in}

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



\vspace{.5in}

7. (a)  (10 pts) Find the eigenvalues and the eigenvectors of the following
matrix : $\mathbf{A=}\left[
\begin{array}
[c]{ccc}%
2 & 2 & 0\\
0 & 0 & 1\\
0 & 0 & 2
\end{array}
\right]  $


(b) (5 pts) What are the algebraic and geometric multiplicities of each eigenvalue of $\mathbf{A}$.

(c) (5 pts) Is this matrix diagonalizable?

\newpage

8. (15 pts) Let $\mathbf{A}$ be the matrix $\left[
\begin{array}
[c]{cc}%
1 & 3\\
-2 & 6
\end{array}
\right]  $ . Find a $2\times2$ matrix $\mathbf{C}$ and a diagonal matrix
$\mathbf{D}$ such that $\mathbf{C}^{-1}\mathbf{AC}=\mathbf{D}$.

\vspace{4in}


9. (15 pts) Let $\mathbf{v}=\left[  2,1,2\right]  $ and let $W=span\left(
\left[  0, 1 , 1 \right] \  ,\ \left[  1,1,0\right]  \right)  $. Find the
orthogonal decomposition $\mathbf{v}=\mathbf{v}_{W}+\mathbf{v}_{\perp}$ of
$\mathbf{v}$ with respect to the subspace $W$.

\newpage

10. (10 pts) Find an orthonormal basis for the subspace $W$ generated by the
vectors $\mathbf{v}_{1}=\left[  1,1,1\right]  $ and $\mathbf{v}_{2}=\left[
1,0,1\right]  $





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