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\title[Linear Algebra]
{Final Exam \\Math 3013 Section 002, Spring 2009}
 
\author[Weiping Li]{Instructor: Weiping Li}
 
\maketitle

{\bf Print Name and Student $\#$}
\vspace{4ex}
\medskip

{\bf SHOW WORK FOR CREDIT !!! SHOW WORK FOR CREDIT !!!}
\medskip

\begin{enumerate}
\item (10pts) (a) Verify the matrix $A= \frac{1}{2}  \left( \begin{array}{cccc} 
1&-1&1&1\\ -1&1&1&1\\ 1&1&-1&1\\1&1&1&-1 \end{array}\right)$ is orthogonal.
\vspace{1.5in}

(b) Let $u = q_1 -2q_2 +2q_3$ for an orthonormal basis $\{q_1, q_2, q_3\}$ in $R^3$. Find $\|u\|$.
\vspace{1in}

\item (10pts) Let $A$ be a $4\times 4$ matrix with $\det A = 5$.  Find the following.

(i) $\det (2 A^T + A^T)$
\vspace{0.5in}

(ii) $\det (A^T \cdot A)$
\vspace{0.5in}

(iii) $\det (A^{-3})$.
\newpage

\item (10pts) Find a basis for the orthogonal complement $W^{\perp}$ in ${\mathbb R}^4$ of the
subspace $W=sp \{ (1, 2, 2, 1), (0, 1, 2, -1)\}$.
\vspace{2in}

\item (15pts) Find an orthogonal basis from the basis $\{(1, 0, 1), (0, 1, 2), (2, 1, 0)\}$ for ${\mathbb R}^3$
by using the Gram-Schmidt process.
\newpage

\item (10pts) Given an orthonormal basis for $W=sp \{v_1, v_2, v_3\}$ in $R^4$, where
$v_1=(1,1,1,1), v_2=(-1,1, -1,1)$ and $v_3=(1, -1,-1,1)$. Find the projection of $b=(1,2,-1,-2)$ on $W$.
\vspace{2.5in}

\item (10pts) Let u and v be vectors in an inner-product space with $\|u\|=3, \|v\|=5$.
If $u$ and $v$ are perpendicular, then find $\langle u+2v, 3u+v\rangle$.
\vspace{2in}

\item (10pts) Determine all values $b_1, b_2, b_3$ such that $b=(b_1,
b_2, b_3)$ lies in the span of ${\bf v}_1= (1, 0, 0), {\bf v}_2 = (1, 2, 1)$ and ${\bf v}_3 = (-1, -2, -1)$.\newpage

\item (15pts) (a) Determine whether the matrix $A = \left(\begin{array}{cc}
2&5\\1&3 \end{array} \right)$ is invertible, and find its inverse.
\vspace{2in}

(b) Express $A$ as a product of elementary matrices.
\vspace{1.5in}

\item (10pts) Find the coordinate vector of the polynomial $x^3-4x^2+3x+7$ relative to the ordered basis
$B_1=\{(x-2)^3, (x-2)^2, (x-2), 1\}$.
\newpage

\item (15pts) Find (i) the rank of $A$ and the nullity of $A$, (ii) a basis for the row space, (iii) a basis for the column space of $A = \left(\begin{array}{ccccc}
1&3&0&-1&2\\ 0&-2&4&-2&0\\ 3&11&-4&-1&6\\ 2&5&3&-4&0 \end{array}\right).$
\vspace{3.5in}

\item (15pts) Let $T$ be a linear transformation such that $T(-1,1) = (2,1,4)$ and $T(1,1) = (-6,3,2)$.

(i) Find $T(x, y)$.
\vspace{2in}

(ii) Using (i),  compute $T(-1, 1)$.
\newpage

\item (15pts) Let $T: P_3 \to P_3$ be defined by $T(p(x)) = (x+2)p^{'}(x)$, where $p^{'}(x)$ is the derivative of $p(x)$. Let $B=\{x^3, x^2, x, 1\}= B^{'}$ be an ordered basis for $P_3$.

(i) Find the matrix representation $A$ of $T$ relative to $B, B^{'}$.
\vspace{2.5in}

(ii) Use $A$ to compute $T(x^3-2x^2+3x-4)$.
\vspace{1in}


\item (10pts)
Find the $x_2$ {\bf ONLY} by {\bf using Cramer's rule} for 
\begin{eqnarray*}
5x_1 - 2x_2 + x_3 & = & 1\\
x_2+x_3 & = & 0\\
x_1+6x_2-x_3 & = & 4.
\end{eqnarray*}
\newpage

\item (10pts)  Find the characteristic polynomial of $\left( \begin{array}{ccc}
-1&0&1\\ -7&2&5\\3&0&1 \end{array} \right)$.
\vspace{2.5in}


\item (10pts) Let $A = \left( \begin{array}{ccc}
-1& -1& -2\\ 8& -11& -8 \\ -10& 11& 7 \end{array} \right)$. The characteristic polynomial of $A$ is
given by $(\lambda - 1)(\lambda +3)^2$.

(i) Find the algebraic multiplicities of $\lambda_1 = 1$ and $\lambda_2 = -3$.
\vspace{0.5in}

(ii) Find the geometric multiplicities of $\lambda_1 = 1$ and $\lambda_2 = -3$.
\vspace{3in}

(i) Is the matrix $A$ diagonalizable ? Must provide a reason for credit!
\newpage


\item (10pts) A matrix $A=\left( \begin{array}{ccc} -3&10&-6\\0&7&-6\\0&0&1 \end{array} \right)$ has 
eigenvalues $\lambda_1=-3$ with eigenvector $v_1 = (1, 0, 0)$, $\lambda_2=1$ with $v_2=(1,1,1)$ and
$\lambda_3=7$ with $v_3=(1,1,0)$.
Compute $A^{2009}$.
\vspace{3in}

\item (15pts) Solve the system of linear differential equations:
\begin{eqnarray*}
x_1^{'} & = &x_1-3x_2+3x_3\\
x_2^{'} & = & -5x_2+6x_3 \\
x_3^{'} & = & -3x_2+4x_3,
\end{eqnarray*}
where the matrix $\left( \begin{array}{ccc} 1&-3&3\\0&-5&6\\0&-3&4 \end{array} \right)$ has eigenvalues $1, 1, -2$.

\end{enumerate} 
\end{document}