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\begin{document}

\leftline{\Large \bf Department of Mathematical Sciences} 
\vspace{12pt}
\leftline{ 21--241 \bf Matrix Algebra \hfill{\bf  Fall 2002}}
\leftline{ \bf Professor A. Frieze and  Professor  S. Ta'asan}


\vspace{1in}
\centerline{\bf Test No. 1}
\vspace{1in}

\leftline{\hspace{3.5in} Name:\hrulefill }
\leftline{\hspace{3.5in} Section:\hrulefill }
\bigskip
\bigskip


\vspace{1.0in}
\hbox{ \hspace{2.0in}
{ \large
\begin{tabular}{||c|c|r||}
\hline
 problem &  points & scores \\ \hline
1 & 25  &\\ \hline
2 & 25 &\\ \hline
3 & 25 &\\ \hline
4 & 25 & \\ \hline
total & 100 & \\
 \hline
\end{tabular}
}
}

\pagebreak

\begin{enumerate}
\item (25 pts) Describe all the solutions to the linear system
$$
 \begin{array}{ccccccc}
x_1 & + & 3 x_2 & - & 5 x_3 & = & 4 \\ 
x_1 & + & 4 x_2 &  - & 8 x_3 & = & 7 \\
-3  x_1 & - & 7 x_2 & + & 9 x_3 & = & -6 
\end{array}
$$
$\left[\begin{array}{rrrr}1&3&-5&4\\1&4&-8&7\\-3&-7&9&-6\end{array}\right]$
\hspace{1in}4 pts\\

\vspace{-20pt}\hspace{2.5in}only 2 pts if RHS left out

\vspace{.25in}
$\left[\begin{array}{rrrr}1&3&-5&4\\0&1&-3&3\\0&2&-6&6\end{array}\right]$
\hspace{1.5in}4 pts

\vspace{.25in}
$\left[\begin{array}{rrrr}1&0&4&-5\\0&1&-3&3\\0&0&0&0\end{array}\right]$
\hspace{1.5in}4 pts

Solution

$\begin{array}{rrrrr}x_1&=&-5&-&4x_3\\x_2&=&3&+&3x_3\end{array}$

or

$\left[\begin{array}{r}-5\\3\\0\end{array}\right]+x_3\left[\begin{array}{r}-4\\3\\1
\end{array}\right]$

\vspace{-55pt}\hspace{2.5in}10 pts

\vspace{50pt}
3 pt bonus for perfect solution.



\newpage
\item (25 points) Determine by inspection if the given set is linearly
  dependent. Give reasons. 

a. 
$$
\left[ \ba{c} 1 \\ 7 \\ 6 \ea \right], 
\left[ \ba{c} 2 \\ 0 \\ 9 \ea  \right], 
\left[ \ba{c} 3 \\ 1 \\ 5 \ea \right],
\left[ \ba{c} 4 \\ 1 \\ 8 \ea \right] 
$$ 
Linearly dependent. More than 3 vectors in $\R^3$ -- 8 pts.

\vspace{2in} 
b. 
$$
\left[ \ba{c} 2 \\ 3 \\ 5 \ea \right], 
\left[ \ba{c} 0 \\ 0 \\ 0 \ea  \right], 
\left[ \ba{c} 1 \\ 1 \\ 8 \ea \right],
$$ 
Linearly dependent. The set contains a zero vector -- 8 pts

\vspace{2in} 
c. 
$$
\left[ \ba{c} -2 \\ 4 \\ 6 \\ 10 \ea \right], 
\left[ \ba{c} 3 \\ -6 \\ -9 \\ 15  \ea  \right]
$$ 
If dependent, one should be a multiple of the other. Linearly independent -- 9 pts.

\pagebreak


\item (25 points) Let $T$ be a linear transformation from $\R^n
  \rightarrow \R^m$. Show that if ${\bf v}_1, {\bf v}_2, \dots, {\bf v}_p$ are linearly
  dependent 
then so are $T({\bf v}_1), T({\bf v}_2), \dots, T({\bf v}_p)$. 

\vspace{1in}
There exist $x_1,x_2,\ldots,x_p$ not all zero such that $x_1{\bf v}_1+x_2{\bf v}_2+\cdots
+x_p{\bf v}_p={\bf 0}.$\\
5 pts, (3 pts if there is an omission of ``not all zero'').

Thus $T(x_1{\bf v}_1+x_2{\bf v}_2+\cdots+x_p{\bf v}_p)={\bf 0}$
 - 5 pts\\
Thus by linearity (5pts)\\
$x_1T({\bf v}_1)+x_2T({\bf v}_2)+\cdots+x_pT({\bf v}_p))={\bf 0}$ - 5 points\\
and so (5 pts) $T({\bf v}_1), T({\bf v}_2), \dots, T({\bf v}_p)$ are linearly dependent..


\pagebreak 
\item (25 points)Mark each statement true or false. No justification is
needed. 

2.5 pts per correct answer.

\vspace{50pt} 
 

(i) Two matrices are row equivalent if they the same number of rows. {\bf F}
\vspace{12pt} 


(ii) Two linear systems are equivalent if they have the same solution
set. {\bf T}
\vspace{12pt} 


(iii) Whenever a system has a free variable then the solution set
contains infinitely many solutions. {\bf F}
\vspace{12pt} 


(iv)  A linear transformation from $\R^n$ to $\R^n$ 
is completely determined by its effect on any $n$ distinct vectors. {\bf F}
\vspace{12pt} 


(v) A vector ${\bf b}$ is a linear combination of the columns of a
matric $A$ if and only if the equation $A {\bf x}  = {\bf b} $ has at
least one solution. {\bf T}
\vspace{12pt} 


(vi) If the columns of a p time q matrix $A$ span $\R^p$ then the
equation $A {\bf x} = {\bf b}$ is consistent for each ${\bf b}$ n
  $\R^p$.{\bf T} \vspace{12pt}

 
(vii) The equation $A {\bf x}  = {\bf b} $ is homogeneous if the zero
  vector is a solution. {\bf T}
\vspace{12pt} 


(viii) The columns of a matrix $A$ are linearly independent if the
  equation 
$A {\bf x} = {\bf 0} $ has only the trivial solution. {\bf T}
\vspace{12pt} 


(ix) Every linear transformation is a matric transformation. {\bf T}
\vspace{12pt} 


(x) The homogeneous equation is always consistent. {\bf T}

\end{enumerate}
\end{document}









 









