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Midterm

MATH240 BOYLE-M SPRING2013 0101 MID EXAM 1Exam


Department
Mathematics
Course Code
MATH 240
Professor
All
Study Guide
Midterm

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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
4485237
14 14 28 4 3 108
0 0 0 0 10 20
15 15 30 10 25 165
,rref(A) =
112007
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) Dene
A=32
1 3 and D={xR2:p(x1π)2+ (x217)23}.
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=2A3B1.
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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