DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 2213

FINAL EXAMINATION
December 1997
TIME: 3 HOURS

SHOW ALL WORK. YOUR METHOD IS MORE IMPORTANT THAN THE RIGHT ANSWER.

MARKS
1. Let \begin{displaymath}A = \left[ \begin{array}{rrrr}
1 & 2 & -1 & -2\\
-3 & 1 & 3 & 4\\
-3 & 8 & 4 & 2
\end{array} \right]\;. \end{displaymath}
(2)(a) Find a basis for Nul A.
(2)(b) Find a basis for Col A.
(2)(c) Find a basis for Row A.
(2)(d) What is the rank of A?
2. Let ${\displaystyle {\bf b}_1 = \left[ \begin{array}{r}
1\\ -2\\ 1
\end{array} \righ...
...pace*{0.5cm} {\bf y} = \left[ \begin{array}{r}
2\\ -3\\ 6
\end{array} \right] }$, and $B = \{b_1, b_2, b_3\}$.
(3)(a) Prove that B is a basis for $\Bbb R^3$. Quote any results you might use.
(2)(b) Find $[{\bf y}]_B$.
(3)(c) Find a matrix P such that $P \cdot [{\bf x}]_B = {\bf x}$ and a matrix Q such that $[{\bf x}]_B = Q{\bf x}$ for all ${\bf x}$ in $\Bbb R^3$.
(You may use part (c) to do part (b) if you wish.)
3. Let ${\displaystyle A = \left[
\begin{array}{rrr}
1 & 5 & -4\\
2 & -4 & 2\\
-6 & -2 & 4
\end{array} \right]\;. }$
(2)(a) Find elementary matrices $E_1,E_2,\ldots,E_k$ (k is the number of elementary matrices that you need) such that
\begin{displaymath}E_k \ldots E_2E_1A = U,
\end{displaymath}
where U is upper triangular.
(2)(b) Find a lower triangular matrix L, such that A = LU. (U as in part (a)).
(4)4. Why is the following set a subspace of $\Bbb R^4$? Find a basis for it.
\begin{displaymath}\left\{ \left[ \begin{array}{c}
a + b\\
2a\\
3a - b\\
-b
\...
...y} \right]\;\;;\;\;\;\;a, b\;\;\mbox{in}\;\;\Bbb R
\right\}\;. \end{displaymath}
(4)5. Let \begin{displaymath}A = \left[ \begin{array}{rrr}
4 & 2 & 3\\
-1 & 1 & -3\\
2 & 4 & 9
\end{array} \right]\;. \end{displaymath}
Show that $\lambda = 3$ is an eigenvalue for A and find a basis for the corresponding eigenspace.
(4)6. Orthogonally diagonalize the matrix ${\displaystyle \left[ \begin{array}{rr}
3 & 1\\
1 & 3
\end{array} \right]\;.}$ What easily checked property of this matrix guarantees that it is orthogonally diagonalizable?
(4)7. Find an orthonormal basis for the subspace of $\Bbb R^4$ spanned by
\begin{displaymath}\left[ \begin{array}{r}
1\\ -4\\ 0\\ 1
\end{array} \right] \h...
...
\left[ \begin{array}{r}
7\\ -7\\ 4\\ 1
\end{array} \right]\;. \end{displaymath}
(4)8. Let \begin{displaymath}A = \left[ \begin{array}{rr}
1 & 5\\
3 & 1\\
-2 & 4
\end{ar...
...} = \left[ \begin{array}{r}
4\\ -2\\ -3
\end{array} \right]\;. \end{displaymath}
Show that $\bf b$ is not in Col (A) and find a least squares solution of $A {\bf x} = b$.
(4)9. Let $B = \{ {\bf b}_1,{\bf b}_2\}$ and $C = \{
{\bf c}_1,{\bf c}_2\}$ be bases for a 2 dimensional subspace of $\Bbb
R^{100}$. Suppose ${\bf b}_1 = -4{\bf c}_1 + 9{\bf c}_2$ and ${\bf
b}_2 = -2{\bf c}_1 + 4{\bf c}_2$. Find the change of coordinate matrices $\stackrel {\textstyle P}{C \leftarrow B}$ and $\stackrel{\textstyle P}{B \leftarrow C}$.
(4)10. Let T be a linear transformation from $\Bbb
R^5$ onto $\Bbb
R^5$. If $\{{\bf v}_1,{\bf v}_2,{\bf v}_3\}$ are independent vectors in $\Bbb
R^5$, prove that $\{ T{\bf v}_1, T{\bf v}_2, T{\bf v}_3\}$ are also independent vectors in $\Bbb R^3$.
(2)11. Let A be a $8 \times 5$ matrix. If null(A) has dimension 2, what are the dimensions of the row and column spaces of A?
(4)12. Suppose ${\bf p}$ is a solution to the system of equations $A{\bf x} = {\bf b}$ (i.e. $A{\bf p} = {\bf b}$). Prove that any solution to $A{\bf x} = {\bf b}$ can be written ${\bf v} + {\bf p}$ where ${\bf v}$ is a solution to $A{\bf x} = {\bf0}$.
13. A is an m by n matrix of rank r. Suppose $A{\bf x} = {\bf b}$ has no solution for some right sides $\bf b$ and infinitely many solutions for some other right sides $\bf b$.
(2)(a) Decide whether the nullspace of A contains only the zero vector and why.
(2)(b) Decide whether the column space of A is all of $\Bbb R^m$ and why.
(2)(c) Can there be a right side $\bf b$ for which $A{\bf x} = {\bf b}$ has exactly one solution? Why or why not?

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