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\noindent
Math 261
\hfill
December 16, 2002

\begin{center}
  {\bf Final}
\end{center}

\bigskip
The total number of points possible on this exam is 175.  165 will
be considered a perfect score.

\bigskip

\begin{enumerate}

\item (15 points)
\begin{enumerate}
\item 
Show that the system 
 \begin{displaymath}
   \begin{array}{rcrcrcr}
    x&+&y&+&2 z & =&  2\\
      3x&+&y&+&3z &  =& 2\\
     7x&-&y&+&az & = & -2
    \end{array}
 \end{displaymath}
has a unique solution if 
\begin{math}
  a\ne 2.
\end{math}
\medskip

\item Find all solutions when 
  \begin{math}
    a=2.
  \end{math}

\medskip
\item List at least three ways you could solve the system if 
  \begin{math}
    a=2002.
  \end{math}
\end{enumerate}

\item (23 points) Let 
  \begin{math}
    T
  \end{math}
be the linear transformation from 
\begin{math}
  \mathbf R^6
\end{math}
to 
\begin{math}
  \mathbf R^4
\end{math}
given by 
\begin{math}
  TX=AX
\end{math}
where
\begin{displaymath}
  A=\left[
    \begin{array}{rrrrrr}
      1&1&0&-7&0&-1\\
      0&1&1&23&-2&0\\
      0&0&0&0&1&-6\\
      0&0&0&0&-1&-7
    \end{array}
    \right]
\end{displaymath}
and 
\begin{math}
  X=(x_1,\dots,x_6)^t.
\end{math}

\begin{enumerate}
\item Find a basis for the null space of 
  \begin{math}
    T.
  \end{math}

\item  Find a basis for the range of T.

\item Find  the null space of 
  \begin{math}
    T^t.
  \end{math}


\item Find a basis for the range of
  \begin{math}
    T^t.
  \end{math}


\item Find the annihilator of the range of T.    

\item What are the rank and nullity of 
  \begin{math}
    T?
  \end{math}
 
\item What are the rank and nullity of
 \begin{math}
    T^t?
  \end{math} 



\end{enumerate}
\item (12 points)
Let 
\begin{math}
  V_n
\end{math}
be the space of real polynomials of degree at most 
\begin{math}
  n.
\end{math}
Let 
\begin{math}
k
\end{math}
be a fixed positive integer, 
\begin{math}
  k<n.
\end{math}
For each of the following subsets of 
\begin{math}
  V_n,
\end{math}
determine whether it is a subspace and if so compute its dimension and
find a basis for it.

\medskip

\begin{enumerate}
\item 
\begin{math}
   S = \{ p\in V_n: p \mbox{\ has degree}\  k\}.
\end{math}

\medskip


\item
 \begin{math}
    S = \{ p\in V_n: p \mbox{\ has degree }\le k\}.
 \end{math}

\medskip

\item
\begin{math}
   S = \{ p\in V_n: p(x) \le 0, \ \mbox{all\ }x\}.
\end{math}

\medskip

\item 
  \begin{math}
    S = \{ p\in V_n: p(0)+3 p''(0)=0\}.
  \end{math}
\end{enumerate}

\bigskip

\item (10 points)
If 
\begin{math}
  V
\end{math}
and 
\begin{math}
  W
\end{math}
are finite dimensional vector spaces and 
\begin{math}
  T:V\to W
\end{math}
is an isomorphism,
prove that 
\begin{math}
  T
\end{math}
maps a basis of 
\begin{math}
  V
\end{math}
onto a basis of 
\begin{math}
  W.
\end{math}

\item (10 points)
Show that the matrices
\begin{math}
  \left[
  \begin{array}{rr}
    1&0\\
    5 & 11
  \end{array}
 \right]
\mbox{ and }
  \left[
\begin{array}{rr}
1 & 11\\
0 & 5
\end{array}
  \right]
\end{math}
are not similar.


\item (10 points)
\label{det}
Use properties of the determinant to evaluate 
\begin{math}
  \left[ 
  \begin{array}{rrrr}
   1 & -2 & 3 & -12 \\
-5 & 12 & -14 & 19 \\
-9 & 22 & -20 & 31 \\
-4 & 9 & -14 & 15
  \end{array}
   \right].
\end{math}

\bigskip 
\item (10 points)
Let 
\begin{math}
  A=\left[ 
    \begin{array}{rr}
      A_{11}& A_{12 }\\
      A_{21}& A_{22}
    \end{array}
   \right].
\end{math}
Find
\begin{math}
  \mbox{adj}\, A.
\end{math}
\end{enumerate}



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