Assignment 3
• Write or type your solution in separate sheets of paper, and
CLEARLY indicate your name and student ID on the TOP of
the FIRST PAGE.
• Show your computations, or explain your answer clearly.
• You may use any result (theorem, property, formula etc.) covered
in lectures.
• Submit your assignment to the MAT1010 dropbox on the sixth floor
of the Cheng Dao Building, before 6PM of Tuesday, April 18.
1. Consider the following vectors:
 
 
 
 
 
 
4
0
0
1
1
2
v~1 = 6 , v~2 = 1 , v~3 = 1 , u~1 = 1 , u~2 = 2 , u~3 = 3 .
7
1
2
1
2
4
Given that B1 = {v~1 , v~2 , v~3 } and B2 = {u~1 , u~2 , u~3 } are both bases for R3 :
(a) Find the transition matrix from B1 to B2 .
(b) If ~x = 2v~1 + 3v~2 − 4v~3 , determine the coordinates of ~x with respect to
B2 .
2. For each of the following functions L, determine whether L is a linear
transformation or not. Justify your answer.
x1
x1
2
2
(a) L : R → R , L
=
.
x2
sin x2


x1 − 9x2
x
(b) L : R2 → R3 , L 1 =  −6x1 .
x2
48x2
1
3. Consider the linear transformation L : R2 → R2 that first rotates a vector
clockwise around the origin by an angle of 32 π and then reflects it about
the x2 -axis.
(a) Find the standard matrix of L.
(b) Find the range and null space of L.
4. Let L : R2 → R3 be a linear transformation satisfying
 
 
1
2
1
2
L
=  6  and L
=  8 .
3
4
15
22
Determine the standard matrix of L.
5. Compute the determinant of each of the following matrices:


3 2 4
(a) 1 −2 3
2 3 2


1
3 −2 0
5
−2 −6 4
4 −7



0
3 −1 1 
(b)  0

0
2 −5 1
4
−1 −3 2
0
0
6. You are given that
1
0
2
3
2
0
3
6
3
4
5
9
4
8
= −48.
7
0
Without using cofactor expansion, determine
1
0
2
3
3
0
5
9
4 6
8 8
.
7 10
0 18
Explain why your answer is correct. (You may only use results covered in
lectures.)
2
7. For each of the following statements, determine whether it is true or false.
Justify your answer in each case.
(a) If A and B are 3 × 3 matrices, then det(A + B) = det(A) + det(B).
(b) If A is a 2 × 2 matrix and B = 2A, then det(B) = 2 det(A).
(c) If A is an n×n matrix, then the columns of A are linearly independent
if and only if the rows of A are linearly independent.
3