MATH 221 - 102
EXAM #2
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Student ID:
Exam rules:
• No calculators, open books or notes are allowed.
• There are 5 problems in this exam. All problems are worth the same number
of marks.
• Some problem contain several parts. Please read the problems carefully.
• Use opposite empty pages if needed.
1
Problem 1. Let R : R2 → R2 be the rotation by 60◦ angle counterclockwise, and
let S be the transformation
¸
· ¸ ·
2x1 − 3x2
x1
.
=
S
−x1 + 2x2
x2
Find the standard matrix of the composition T :
T (~x) = R(S(~x)).
Problem 2. Let


0 1
2 −3 7
1 1
3 −3 6 

A=
1 −1 −1 3 −7 .
2 0
2
0
2
(a)Find a basis for the nullspace of A and a basis for the column space of A.
(b)Determine if the linear transformation defined by A is one-to-one or onto?
Problem 3. Let


1 −2 2
A = −2 5 −6
1 −4 5
(a)Find the inverse of A.
(b)Find the inverse of AT · A.
Problem 4. Find the determinant of the matrix ( 12 A)−1 , where


1 2 −3 4
−4 2 1
3

A=
 3 0 0 −3 .
2 0 −2 3
Problem 5. Answer each question ”True” or ”False”. No explanation is necessary.
(a)If AB + BA = 0, then A2 B 3 = B 3 A2 , where A, B are n × n matrices.
(b)If A is a 5 × 2 matrix then the transformation S(~x) = AT A~x is linear.
(c)If a linear transformation T has standard matrix A of size 4 × 5, then T can
not be onto.
(d)If AP = P B for some invertible matrix P , then A and B have the same determinant.
(e)When A, B are n × n matrices, then the set of all vectors ~v such that A~v = B~v
is a subspace of Rn .
(f)If A is a 6 × 4 matrix such that its nullspace has dimension 2, then the linear
transformation defined by A is onto.