DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 2213

FINAL EXAMINATION
APRIL 1999
TIME: 3 HOURS

ANSWER ALL QUESTIONS WITHOUT USE OF A CALCULATOR. CREDIT WILL BE GIVEN FOR METHOD AND PRESENTATION OF SOLUTIONS.

MARKS
(8) 1. Row reduce the augmented matrix,
\begin{displaymath}[A,{\bf b}]= \left[ \begin{array}{rrr\vert r}
1 & 4 & -2 & 1\...
...-1 & 2\\
5 & -8 & 2 & 5\\
1 & -3 & 1 & 1
\end{array} \right] \end{displaymath}
to an echelon form and, hence, determine all solutions of the linear system Ax = b.
(8) 2. Let ${\displaystyle
A = \left[ \begin{array}{rrr}
2 & 4 & 2\\
-1 & 1 & 0\\
4 & -1 & 4
\end{array} \right]\;.}$
(a) Compute detA.
(b) Use elementary row operations to determine the inverse of A.
(c) Factor A as a product LU, wherein L denotes a lower triangular matrix with unit diagonal entries and U denotes an upper triangular matrix.
(9)3.(a) Given ${\displaystyle {\bf u} = \left[
\begin{array}{r}
1\\ -1\\ 2
\end{array} \right]}$ and ${\displaystyle {\bf v} = \left[
\begin{array}{r}
-2\\ 1\\ 0
\end{array} \right]}$, determine whether or not the vector ${\displaystyle {\bf w} = \left[ \begin{array}{r}
1\\ -2\\ 3
\end{array} \right]}$ belongs to the subspace, span $\{{\bf u,v}\}$, in $\Bbb R^3$.
(b) Find a basis for the column space of the matrix,
\begin{displaymath}A = \left[ \begin{array}{rrrr}
2 & 1 & 5 & -1\\
1 & 3 & 5 & 0\\
-1 & 0 & -2 & 3
\end{array} \right]\;. \end{displaymath}
Justify your answer.
(c) Let $\{{\bf v}_1,{\bf v}_2,{\bf v}_3\}$ be three non-zero vectors from $\Bbb R^3$. Prove that if the vectors are mutually orthogonal then they are linearly independent.
(10)4.(a) Let ${\cal B} = \{{\bf b}_1,{\bf b}_2\}$ and ${\cal C} = \{{\bf
c}_1,{\bf c}_2\}$ be two bases of $\Bbb R^2$ in which
\begin{displaymath}{\bf b}_1 = \left[ \begin{array}{c}
1\\ 0
\end{array} \right]...
...f c}_2 = \left[ \begin{array}{r}
1\\ -2
\end{array} \right]\;. \end{displaymath}
(i) Determine the change of coordinates matrix, P, for the change ${\cal B}$ to ${\cal C}$.
(ii) Given the coordinates, ${\displaystyle [{\bf x}]_{\cal B} =
\left[ \begin{array}{c}
3\\ 2
\end{array} \right]}$, for a vector ${\bf x}$ relative to ${\cal B}$, find $[{\bf x}]_{\cal C}$.
(b) Determine an orthogonal base for the subspace $S =
\mbox{span}\{{\bf u},{\bf v}\}$ in which ${\displaystyle
{\bf u} = \left[ \begin{array}{r}
1\\ -1\\ 1
\end{array} \right]\;,\;\;{\bf v} = \left[ \begin{array}{r}
2\\ 0\\ -1
\end{array} \right] .}$
(10)5.(a) Let $T:\Bbb R^3 \rightarrow \Bbb R^2$ be a linear mapping for which
\begin{displaymath}T({\bf e}_1) = \left[ \begin{array}{c}
1\\ 4
\end{array} \rig...
...\bf e}_3) = \left[ \begin{array}{r}
3\\ -8
\end{array} \right] \end{displaymath}
where {e1, e2, e3} is the standard basis for $\Bbb R^3$.
(i) Is T a one-to-one mapping? (ii)   Is T an onto mapping?
Justify your answers to these questions.
(b) Let $T:\Bbb R^2 \rightarrow \Bbb R^2$ be the linear transformation for which ${\displaystyle T({\bf x}) = \left[
\begin{array}{rr}
2 & -1\\
1 & 3
\end{array} \right]{\bf x}}$. If ${\cal B} = \{{\bf b}_1,{\bf b}_2\}$ where ${\displaystyle
{\bf b}_1 = \left[ \begin{array}{r}
-1\\ 2
\end{array} \right],\;\;\;{\bf b}_2 = \left[ \begin{array}{c}
1\\ 1
\end{array} \right]}$ calculate the matrix, $[T]_{\cal B}$, for T relative to base ${\cal B}$.
(8)6. Find the eigenvalues and eigenvectors of the matrix ${\displaystyle A = \left[ \begin{array}{rrr}
-1 & 1 & 1\\
1 & 1 & 0\\
-1 & 0 & 1
\end{array} \right]}$ .
(10)7. The vectors ${\displaystyle {\bf x}_1 = \left[
\begin{array}{r}
2\\ 2\\ -1
\end{array} \righ...
...ght]\;,\;\;{\bf x}_3 = \left[ \begin{array}{r}
2\\ -1\\ 2
\end{array} \right] }$ constitute a linearly independent set of eigenvectors of the symmetric matrix,
\begin{displaymath}A = \left[ \begin{array}{rrr}
7 & -4 & 4\\
-4 & 5 & 0\\
4 & 0 & 9
\end{array} \right] . \end{displaymath}
(a) Verify that the eigenvectors are mutually orthogonal.
(b) Construct an orthogonal matrix P with which A is orthogonally diagonalizable as A = PDP-1 for some diagonal matrix D.
(c) Find (i) the matrix P-1, and (ii) the matrix D.

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