Math 46

Final Exam

Spring 2006 Your name

Directions: For each problem, place the letter choice of your answer in the spaces provided on this page.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

By signing here, I pledge that I have neither given nor received assistance on this exam.

Your signature

1

1. Given that A∼B, where

A=

1 4 0 2 −1

3 12 1 5 5

2 8 1 3 2

5 20 2 8 8

and B=

140 20

001−1 0

000 01

000 00

,

the dimension of Nul(A) is

(a) 1 (b) 2 (c) 3 (d) 4 (e) 5

2. Suppose v1,v2,v3,v4,v5∈Vand 2v1+ 3v3−v4=0.

I. v1,v3,v4are linearly dependent.

II. v1,v2,v3,v4,v5are linearly dependent.

III. v2,v5are linearly independent.

(a) Only I is true

(b) Only II is true

(c) I and II are true

(d) Only III is true

(e) I and III are true

3. Let U= x

y∈IR2|x+ 2y= 0. Let V= x

y∈IR2|y= 2x2.

(a) Only Uis a subspace of IR2

(b) Only Vis a subspace of IR2

(c) Both Uand Vare subspaces of IR2

(d) Neither Unor Vis a subspace of IR2

4. Let T: IR3→IR5be a linear transformation with T

1

2

3

=

0

0

0

0

0

. Then

(a) Tis 1-1 but not onto

(b) Tis onto but not 1-1

(c) Tis 1-1 and onto

(d) Tis neither 1-1 nor onto

(e) There is not enough information to decide

5. Suppose Ais 10 ×12 and T(x) = Ax. Suppose dim(ker(T)) = 3.

ITis not 1-1

II. Tis not onto

(a) Only I is true

(b) Only II is true

(c) I and II are both true

(d) There’s not enough information to decide

2

## Document Summary

Directions: for each problem, place the letter choice of your answer in the spaces provided on this page. By signing here, i pledge that i have neither given nor received assistance on this exam. 1: given that a b, where. Ii. v1, v2, v3, v4, v5 are linearly dependent. T is not onto (a) only i is true (b) only ii is true (c) i and ii are both true (d) there"s not enough information to decide. 3: warning: look at the elements of s very carefully. x y z. 7 (b) 9 (c) 16 (d) 63: let a and b be matrices. Suppose t1 : ir3 ir5 is given by t1(x) = ax and t2 : ir2 ir3 is given by. 1: for the eigenvalue = 1 of the matrix a = . , the eigenspace is k-dimensional, where k = (a) 1 (b) 2 (c) 3 (d) 1 is not an eigenvalue of a.