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,
Ax = b.
| ||

| Let | |||

(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.
| |||

Given and , determine whether or not the vector belongs to the subspace, span , in . | ||||

(b) | Find a basis for the column space of the matrix,
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Let be three non-zero vectors from . Prove that if the vectors are mutually orthogonal then they are linearly independent. | ||||

Let
and
be two bases of
in which
| ||||

(i) | Determine the change of coordinates matrix, P, for the change
to .
| |||

Given the coordinates,
,
for a vector
relative to ,
find
. | ||||

(b) | Determine an orthogonal base for the subspace in which | |||

Let
be a linear mapping for which
e_{1}, e_{2}, e_{3}}
is the standard basis for
.
| ||||

(i) | Is T a one-to-one mapping? T an onto mapping?
| |||

Justify your answers to these questions. | ||||

Let
be the linear
transformation for which
.
If
where
calculate the matrix,
,
for T
relative to base .
| ||||

Find the eigenvalues and eigenvectors of the matrix . | ||||

The vectors
constitute a linearly independent set of
eigenvectors of the symmetric matrix,
| ||||

(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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(70) |