Precise Description - Applied Linear Algebra - Exam, Exams of Linear Algebra

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SIMON FRASER UNIVERSITY

DEPARTMENT OF MATHEMATICS AND STATISTICS

Final Exam

MATH 232

December 15, 1999, 3:30 – 6:30 p.m.

Name: family name given name

Number:

INSTRUCTIONS

1. This exam has 14 questions on 16 pages. Please check to make sure your exam is complete.
2. Write your final answer in the answer box.
3. In each question indicate how you obtain your answer. You may lose points if your work is poorly presented.
4. If you need more room, use the reverse side of the previous page to show your work.
5. No calculators or other computing devices may be used.
6. Please write with a black or blue pen.

Question Score Max

Total 100

[6] 2. Find a basis for the set of solutions to the

system

2 x 1 − x 2 − 6 x 3 + 10 x 4 = 0 −x 1 + 3 x 2 + 8 x 3 − 15 x 4 = 0.

ANSWER

SHOW YOUR WORK

3. Let V = sp(a 1 , a 2 , a 3 , a 4 , a 5 ) denote the subspace of R^5 spanned by

a 1 = [2, 2 , 1 , − 1 , 0] a 2 = [− 1 , 1 , 1 , 2 , 2] a 3 = [7, 1 , − 1 , − 8 , −6] a 4 = [0, 8 , 6 , 6 , 8] a 5 = [1, 1 , 1 , − 1 , −3]

and A =

[

a 1 a 2 a 3 a 4 a 5

]

be the 5 × 5 matrix whose columns are a 1 , a 2 , a 3 , a 4 , a 5.

By elementary row operations A is converted to

H =

[2] (a) Write down a basis for V. ANSWER

[2] (b) Write down a basis for the row space of A.

ANSWER

[2] (c) Determine the rank of A. Give a reason for your an- swer.

ANSWER

ROUGH WORK IF REQUIRED

(use the back of page 2)

[5] 5. Let V be a vector space over R. Let W 1 and W 2 be two subspaces of V. Prove that their

intersection W 1 ∩ W 2 is a subspace of V.

ANSWER

ROUGH WORK

[6] 6. Let R^2 ×^2 denote the vector space of all 2 × 2 real matrices, using as vector addition and scalar

multiplication the usual addition of matrices and multiplication of a matrix by a scalar.

Given are four matrices

v 1 =

[

]

, v 2 =

[

]

, v 3 =

[

]

, v 4 =

[

]

It is given that B = (v 1 , v 2 , v 3 , v 4 ) is an ordered basis for R^2 ×^2.

Let

v =

[

]

Find the coordinate vector vB of v relative to B. ANSWER

SHOW YOUR WORK

(Question 7. continues here.)

[3] (b) Decide whether the transformation T 1 is invertible. Justify your answer.

ANSWER

[2] (c) Use the composition of linear transformations to discover a simple relation between A 2 and A 1. Justify your answer. Do not compute A 2 explicitly, just express it in terms of A 1.

ANSWER

ROUGH WORK IF REQUIRED

[4] 8. (a) Given are three points P = (3, −1), Q = (2, 2) and R = (− 1 , 7).

Find the area of the triangle P QR.

ANSWER

[4] (b) State the row-interchange property for determinants of square matrices. Use it to prove:

If two rows of a square matrix A are equal, then det(A) = 0.

ANSWER

SHOW YOUR WORK

10. Let

A =

[5] (a) Find the eigenvalues and corresponding eigenspaces of A.

ANSWER

SHOW YOUR WORK

(Question 10. continues here.)

[3] (b) Use diagonalization to compute A^2000. Give your answer in the form of a single 3 × 3 matrix.

ANSWER

[3] (c) Decide whether the matrix

B =

[

]

is diagonalizable. Justify your answer.

ANSWER

SHOW YOUR WORK

12. Let V = sp(a 1 , a 2 , a 3 ) be the subspace of R^4 spanned by the vectors a 1 = [1, 0 , 0 , 1], a 2 = [1, 1 , 0 , 1]

and a 3 = [0, 1 , − 1 , 0].

[6] (a) Find an orthogonal basis for V.

ANSWER

[2] (b) Use your answer to part (a) to find an or- thonormal basis for V.

ANSWER

SHOW YOUR WORK (use the back of the previous page if necessary)

[7] 13. The following data points are given:

Find the least-squares linear fit for these data points.

ANSWER

SHOW YOUR WORK