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**preview**shows half of the first page. to view the full**2 pages of the document.**MATH 240 – Spring 2013 – Exam 2

CALCULATORS ARE NOT ALLOWED

TURN OFF ALL ELECTRONIC DEVICES

.

*** THERE ARE QUESTIONS ON BOTH SIDES OF THIS PAPER ***

Answer each question on a separate sheet of paper. Use the back side if necessary. On each sheet, put your

name, your section leader’s name and your section meeting time. No proof is needed for TRUE-FALSE

questions; just write clearly. You may assume given matrix expressions are well deﬁned (i.e. the matrix sizes

are compatible).

1. (a) Below are a matrix Aand the matrix rref(A) produced by MATLAB.

A=

22 −22 44 14 7 182

4−485237

14 −14 28 4 3 108

0 0 0 0 10 20

15 −15 30 10 25 165

,rref(A) =

1−12007

0 0 0 1 0 1

0 0 0 0 1 2

0 0 0 0 0 0

0 0 0 0 0 0

.

Write down a basis for each of the following (no justiﬁcation required).

i. (4 points) Row(A), the row space of A.

ii. (4 points) Col(A), the column space of A.

iii. (6 points) Nul(A), the null space of A.

(b) (6 points) For each of the following, answer TRUE or FALSE.

i. rank(A+B)≥rank(A) whenever Aand Bare 10 ×21 matrices.

ii. rank(AB)≤rank(A) whenever Aand Bare 10 ×10 matrices.

2. (a) (15 points) Deﬁne

A=−3−2

1 3 and D={x∈R2:p(x1−π)2+ (x2−17)2≤3}.

Compute the area of the set E={Ax :xis in D}.

(b) (5 points) Suppose Aand Bare 5 ×5 matrices with det(A) = 10 and det(B) = 4 .

Compute the determunant of the matrix M=−2A3B−1.

3. (a) (14 points) Let P1denote the vector space of polynomials of degree at most 1; then

B={3 + t, 5 + 5t}is a basis of B.

Find the coordinates vector x= [−7 + t]Bof the polynomial −7 + t.

(b) (6 points) For each of the following, answer TRUE or FALSE.

i. R2is a subspace of R3.

ii. If Cis a set of 14 vectors which span R5, then Ccontains every basis of R5.

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