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Transcript
Mathematics 220 in-class exercises
Tuesday, July 22, 2014
1. Suppose
right

2
 1
A=
 3
4
that the matrix A on the left row-reduces to the matrix B to the
1
3
2
−1
3
−1
4
9

1 4
1 −2 

1 9 
1 18

1
 0
B=
 0
0
=⇒
reduces to
0
1
0
0
2
−1
0
0

0 0
0 0 

1 0 
0 1
Denote the columns of A by ~a1 , ~a2 , . . ., ~a5 , and the columns of B by ~b1 , ~b2 ,
. . ., ~b5 .
(a) True or False: The solution set of A~x = ~0 is identical to the solution set
of B~x = ~0.
(b) Is ~a3 ∈ span{~a1 , ~a2 } ?
(c) Is span{~a1 , ~a2 } = span{~b1 , ~b2 } ?
(d) Do the columns of A form a linearly independent set?
(e) Is the set {~a3 , ~a4 , ~a5 } a linearly independent set?
If linear transformation T (~x) = A~x,
(f) What is the domain of T ? the codomain of T ?
(g) Is the linear transformation T (~x) = A~x onto its codomain? one-to-one ?
(h) What is the range of T ?
(i) If T (~x) = B~x, is the range of S = range of T ?
2. Suppose that A is a 3 × 3 matrix and
a
det(A) = r
x
Find the values of
(a) det(3A) (b) det(A3 )
that
b
s
y
(c) det((2A)−1 )
c
t
z
=6
(d) det(AT A)
(e) det(P AP −1 )
(in part (e) assume that P is an invertible 3 × 3 matrix)
a − 2x
x
(f) r
b − 2y
y
s
c − 2z
z
t
a
0
(g) 2r
+
3x
x
b
0
2s + 3y
y
1
c
2
0
9 2t + 3z
14
z
(h) Find the value of the matrix entry (A−1 )2,3
(i) If S is the unit ball {(x1 , x2 , x3 ) ∈ R3 : x2 + y 2 + z 2 ≤ 1}, find the volume
of the image of the unit ball under multiplication by the matrix A.