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\begin{center}
Math 3013.62663\\%[0pt]
Final Exam\\%[0pt]
December 10, 2021 : 10:00am - 11:50am
\end{center}

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


1. Complete the following definitions 

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

\vspace{1in}

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

\vspace{1in}

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

\vspace{1in}

(d) (5 pts) A \textbf{linear transformation} between two vector spaces $V$ and
$W$ is $\ldots$

\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]{ccccc|c}%
0 & -1 & 0 & 1 & 2 & 1\\
0 & 0 & 2 & 0 & 1 & 0\\
0 & 0 & 0 & 2 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 0
\end{array}
\right]  $

(b) (5 pts)$\left[
\begin{array}
[c]{cccc|c}%
1 & 0 & 4 & 2 & 1\\
0 & 2 & 0 & 3 & 2\\
0 & 0 & 0 & 0 & 1\\
0 & 0 & 0 & 0 & 0
\end{array}
\right]  $

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

\newpage

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

\vspace{5in}

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

\newpage

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

(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}

6. 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_{1}\,,\,x_{1}-x_{2}\right]  $.

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

\vspace{2.0in}

(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

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

(b) (5 pts) What are the algebraic and geometric multiplicities of each eigenvalue?

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

\newpage

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

\newpage

9. Let $W=span\left(  \left[  1\ ,1,1\right]  ,\ \left[  0,1,1\right]
\right)  \subset\mathbb{R}^{3}$. 

(a) (7 pts) Find the orthogonal complement $W_{\perp}=\left\{  \mathbf{x}%
\in\mathbb{R}^{3}\mid\mathbf{x}\cdot\mathbf{w}=0\text{ for all }\mathbf{x}\in
W\right\}  $ of $W$ in $\mathbb{R}^{3}.$

(b) (8 pts) Let $\mathbf{v}=\left[  1,-2,0\right]  $, find the orthogonal
decomposition $\mathbf{v}=\mathbf{v}_{W}+\mathbf{v}_{\perp}$; $\mathbf{v}%
_{W}\in W$, $\mathbf{v}_{\perp}\in W_{\perp}$, of $\mathbf{v}$ with respect to
$W$.



\vspace{5in}

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


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