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\title[Linear Algebra]
{Exam 1\\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 (5pts) Fill in the missing entries in the $4\times 4$ matrix below so that the matrix is symmetric.
\[ \left( \begin{array}{cccc}
1& -1 &  & 5\\  & 4&  & 8\\  2& -7 & -1& \\ & & 6& 3 \end{array}\right).\]
\vspace{.1in}

\item (15pts) Let ${\bf u} = (1, -1, 0)$ and ${\bf v} =(0, 1, -1)$.

(i) Find the angle between $\bf u$ and $\bf v$; 

(ii) find $\|{\bf u} + 2{\bf v}\|$;

(iii) Find the unit vector parallel to ${\bf u} + 2{\bf v}$, 
having the opposite direction.
\newpage

\item (10pts) Prove that $(2, 0 , 4), (4, 1, -1)$ and $(6, 7, 7)$ are
vertices of a right triangle in ${\mathbb  R}^3$.
\vspace{2.5in}

\item (15pts) Determine whether the vectors $v_1 = (1, 1, -1), v_2=(-3, -2, 1)$ and $v_3=(1, 3, -5)$ form a basis for ${\bf R}^3$.
\vspace{1.8in}

\item (10pts) Let $A=\left(\begin{array} {ccc}
1& 0&-1\\0&1& 1 \end{array} \right)$ and $B= \left(
\begin{array}{cc} 1&0 \\-1&1\\ 0&-1 \end{array} \right)$.
Compute (i) $A\cdot B$; and (ii) $B\cdot A$.
\newpage

\item (15pts) (a) Determine whether the matrix $A = \left(\begin{array}{ccc}
1&3&-2\\ 2&5&-3\\ -3&2&-4 \end{array} \right)$ is invertible, and find its inverse.
\vspace{4in}

(b) Express the invertible matrix $B=\left(\begin{array}{cc}2&9\\1&4 \end{array}\right)$ as a product of elementary matrices.
\newpage

\item (15pts) Find a basis for the solution set of the homogeneous linear system
\begin{eqnarray*}
x_1-x_2+x_3-x+x_4& = &0\\
x_2+x_3& = & 0\\
x_1+x_2-x_3+3x_4& = & 0.
\end{eqnarray*}
\vspace{3in}

\item (15pts) Solve the given system
\[2x_1-x_2+2x_3=-3, \ \ \ \ 2x_1+x_2-2x_4=1.\]
Express your solution as a particular solution with solutions from homogeneous system.
\end{enumerate} 
\end{document}