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   \par\medskip\noindent\theprobno . (#1 points) }
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\begin{document}

\begin{flushright}
  A. Miller\\
  M340 Final Exam\\
  May 14, 1990
\end{flushright}


{\bf There is a total of 35 points.
Put your answers on separate sheets of paper.
Keep this exam, only hand in the answers. You can pick up your
graded exam on Wednesday, VV403, 2-3.
Your grade will be posted on the door. }

\prob{3} Find a diagonal matrix $D$ and invertible matrix $P$ such
that $A=PDP^{-1}$ where
$$
A=
\left[
\begin{array}{rr}
 1 & -2 \\
 1 & 4
\end{array}
\right]
$$

\prob{6} Find bases for the $kernel(A)$ and the $range(A)$ where
$$
A=
\left[
\begin{array}{rrrrr}
         1    &     2    &    -2    &    -3    &     2\\
        -1    &    -2    &     3    &     4    &     0\\
         3    &     6    &    -3    &    -6    &    -3
\end{array}
\right]
$$
\prob{3} Find an orthonormal sequence $u_1,u_2,u_3$ so that
$span\{v_1,\ldots,v_k\}=span\{u_1,\ldots,u_k\}$ for $k=1,2,3$.
$$
v_1=
\left[
\begin{array}{r}
         0  \\
         1  \\
         0
\end{array}
\right]
\;\;
v_2=
\left[
\begin{array}{r}
        -3  \\
         2  \\
         4
\end{array}
\right]
\;\;
v_3=
\left[
\begin{array}{r}
          4  \\
         -8  \\
          3
\end{array}
\right]
$$

\prob{3} Find the inverse of the matrix $A$:
$$
A=
\left[
\begin{array}{rrr}
         1   &     2   &     0   \\
         1   &     3   &     1    \\
        -1   &    -1   &     2
\end{array}
\right]
$$

\prob{3} Find the determinant of the matrix $A$:
$$
A=
\left[
\begin{array}{rrrrr}
          1   &      0    &     0    &     0   &      0  \\
        -17   &      0    &     3    &     0   &    -85  \\
          6   &     -1    &     0    &     0   &      0  \\
         -4   &      0    &     0    &     1   &      0  \\
         12   &      3    &     0    &     0   &     -2
\end{array}
\right]
$$
\prob{4} Suppose that $A\in \field^{n\by m}, B\in \field^{m\by p},
C\in \field^{p\by r}$.  \par Prove that $(AB)C=A(BC)$.

\prob{4}
If $W_1$ and $W_2$ are subspaces of a vector space $V$, then define
$$W_1\bigoplus W_2=\{w_1+w_2:w_1\in W_1,w_2\in W_2\}.$$
Prove that  $W_1\bigoplus W_2$ is a subspace of $V$.

\prob{4} Suppose that $u_1,u_2$ is an orthonormal sequence in an inner
product space and $c_1,c_2\in\reals$.  Prove that
$\norm{c_1u_1+c_2u_2}=\sqrt{c_1^2+c_2^2}$.

\prob{4} Suppose that $U\in \reals^{n\by n}$ is an upper triangular matrix,
$U^T$ is its transpose, and $U^TU=UU^T$.   Prove that $U$ is a diagonal
matrix.

\prob{1} True or False: Suppose that $u_1,u_2,\ldots,u_n$ is an
orthonormal sequence, then $u_1,u_2,\ldots,u_n$ are linearly independent.

%\prob{ } True or False: Every subspace of an inner product space
%has an orthonormal basis.
%
%\prob{ } True or False: If $v_1,v_2,\ldots,v_n$ are linearly independent and
%$v_{n+1}\in span\{v_1,\ldots,v_n\}$, then $v_1,v_2,\ldots,v_n,v_{n+1}$
%are linearly independent.
%
%\prob{ } True or False: Suppose
%$u_1,u_2,\ldots,u_n$ are linearly independent, and
%$u$ is not the zero vector.  Then for some
%$i$ the vectors $$u_1,u_2,\ldots,u_{i-1},u_{i+1},\ldots,u_n, u$$ are
%linearly independent.
%
%\prob{ } True or False: Suppose that $L:V\mapsto V$ is a
%linear transformation and $v_1,v_2,\ldots,v_n$ are linearly independent,
%then $L(v_1),L(v_2),\ldots,L(v_n)$ are linearly independent.


\newpage

\begin{center}
   Answers
\end{center}

\begin{verbatim}
1.  p =                           d =
         1.000   -2.000                 2.000    0.000
        -0.500    2.000                 0.000    3.000
2.  basis of range:          1.000   -2.000    2.000
                            -1.000    3.000    0.000
                             3.000   -3.000   -3.000
    basis of kernel:
        -2.000    1.000
         1.000    0.000
         0.000   -1.000
         0.000    1.000
         0.000    0.000
3.         u1      u2       u3
         0.000   -0.600    0.800
         1.000    0.000    0.000
         0.000    0.800    0.600

4.       7.000   -4.000    2.000
        -3.000    2.000   -1.000
         2.000   -1.000    1.000

5. -6
\end{verbatim}

\newpage

\bigskip

6. $$entry_{ij}((AB)C)=\Sigma_k entry_{ik}(AB)entry_{kj}(C)= $$
$$\Sigma_k\Sigma_l entry_{il}(A)entry_{lk}(B)entry_{kj}(C)$$
$$=\Sigma_lentry_{il}(A)\Sigma_k entry_{lk}(B)entry_{kj}(C)$$
$$=entry_{ij}(A(BC))$$

\bigskip

7. $c_1(w_1+w_2)+c_2(w_1^\prime+w_2^\prime)=
(c_1w_1+c_2w_1^\prime)+(c_1w_2+c_2w_2^\prime)$

\bigskip

8. $\norm{c_1u_1+c_2u_2}^2=\langle c_1u_1+c_2u_2, c_1u_1+c_2u_2\rangle$
use linearity of inner product and $\langle u_1,u_2\rangle=0$ and
$\langle u_1,u_1\rangle=\langle u_2,u_2\rangle=1$.

\bigskip

9. Use that $entry_{ii}(U*U^T)=||row_i(U)||^2$,
$entry_{ii}(U^T*U)=||col_i(U)||^2$
and look at minimal row $i$ with a nonzero offdiagonal entry.

\bigskip

10. True


\end{document}