## [answered] Be sure this exam has 3 pages. THE UNIVERSITY OF BRITISH CO

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Be sure this exam has 3 pages.

THE UNIVERSITY OF BRITISH COLUMBIA

Sessional Examination - December 2008

MATH 223: Linear Algebra

Instructor: Dr. R. Anstee, section 101

Special Instructions: No Aids. No calculators or cellphones.

You must show your work and explain your answers. time: 3 hours 1. [15 marks] Consider the matrix equation Ax = b with

1

2

A=

1

1 1

3

2

3 2

4

2

2 0

1

1

2 ?1

?1

0

2 1

3

,

2

3 2

7

b= 5

9 There is an invertible matrix M so that

1

0

MA = 0

0 1

1

0

0 2

0

0

0 0 ?1

1 1

0 1

0 0 1

1

,

0

0 2

3

Mb = 1

0 a) [2 marks] What is rank(A)?

b) [4 marks] Give the vector parametric form for the set of solutions to Ax = b.

c) [6 marks] Give a basis for the row space of A. Give a basis for the column space

of A. Give a basis for the null space of A.

d) [2 marks] Let A0 be the 4 ? 5 matrix obtained by deleting the 5th column of A

from A. What is the rank of A0 ?

2. [15 marks] Let 3 2 A= 2 0

2 1 2

1

0 Determine an orthonormal basis of eigenvectors and hence an orthogonal matrix Q

and a diagonal matrix D so that A = QDQT . You may find it useful to know that 5

is an eigenvalue of A.

3. [7 marks] Determine the matrix A corresponding to the linear transformation from

R3 to R3 of projection onto the vector (1, 2, 3)T . 1 MATH 223 Final Exam 2008 page 2 4. [8 marks] Consider the 2 ? 2 matrix A as follows

 ?4

A=

?12 



 

?3

4 0

1 2

4

=

2

0 2

2 3

10 

2

.

?1 Define an , bn , cn , dn using

 an

A =

cn

n Compute an

,

n?? bn

lim bn

dn  cn

n?? dn

lim 5. [10 marks] You are attempting to solve for x, y, z in the matrix equation Ax = b

where 1 ?1 1

3

x

2 1 ?1 ?1 A=

, x = y, b = 1 1 ?1

1

z

1 1

1

0

? in the column space of A (and hence with ||b ? b||

? 2

Find a ?least squares? choice b

? for x, y, z.

being minimized) and then solve the new system Ax = b

d

6. [15 marks] The differentiation operator ? dx

? maps (differentiable) functions into functions. The operator can be viewed as a linear transformation on the vector space of

differentiable functions. Consider the 3-dimensional vector space P2 of all polynomials in x of degree at most 2. Then two possible bases for P2 are V = {1, x, x2} and

U = {1 + x, x + x2 , x2 + 1}.

d

a) [5 marks] Give the 3?3 matrix A representing the linear transformation dx

acting

on P2 with respect to the basis V .

d

with respect to the basis U . You

b) [5 marks] Give the matrix B representing dx

may find it helpful to note that 1

x

x2 =

=

= 1

(1 + x)

2

1

2 (1 + x)

? 21 (1 + x) c) [5 marks] Is matrix A diagonalizable? 2 ? 21 (x + x2 ) + 12 (x2 + 1)

+ 21 (x + x2 ) ? 12 (x2 + 1)

+ 21 (x + x2 ) + 12 (x2 + 1) MATH 223 Final Exam 2008

page 3

7. [10 marks] Let V be a finite dimensional vector space and assume X = {x1 , x2 , . . . , xk }

is a linearly independent set of k vectors and assume Y = {y1 , y2 , . . . , yk , yk+1 } is a

linearly independent set of k + 1 vectors. Then show that there is some vector in Y ,

say yj , so that {x1 , x2 , . . . , xk , yj } is a linearly independent set of k + 1 vectors.

8. [10 marks] For what values of k is the following matrix diagonalizable? 2 A= k

2 0

1

?2 0

1

4 Hint: determine eigenvalues for A. What is required to make A diagonalizable?

9. [10 marks]

a) [4 marks] Let {v1 , v2 , v3 } be an orthonormal basis for R3 . For any v ? R3 , if

v = c1 v1 + c2 v2 + c2 v3 then show that vT v = ||v||2 = c21 + c22 + c23 .

b) [6 marks] Let A be a symmetric 3 ? 3 matrix with eigenvalues ?1 &gt; ?2 &gt; ?3 .

Show that

?1 = max xT Ax

x where the maximum is taken over all vectors x ? R3 with xT x = 1.

100 Total marks 3

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