Linear Algebra: A Modern Introduction, 4th Edition solutions manual David Poole

Linear Algebra: A Modern Introduction, 4th Edition solutions manual David Poole

 

Chapter 2
Section 2.1
2.1.1 Not a linear transformation, since y2= x2+ 2 is not linear in our sense.
2.1.2 Linear, with matrix


0
2
0
0
0
3
1
0
0


2.1.3 Not linear, since y2= x1x3is nonlinear.
2.1.4 A =



9
3
−3
2
−9
1
4
−9
−2
5
1
5



2.1.5 By Theorem 2.1.2, the three columns of the 2 × 3 matrix A are T(~e1),T(~e2), and T(~e3), so that
A =
?
7
6
−13
11
9
17
?
.
2.1.6 Note that x1


1
2
3

+ x2


4
5
6

=


1
4
2
5
3
6


?x1
x2
?
, so that T is indeed linear, with matrix


1
4
2
5
3
6

.
2.1.7 Note that x1~v1+ ··· + xm~vm= [~v1...~vm]


x1
···
xm

, so that T is indeed linear, with matrix [~v1~v2 ··· ~vm].
2.1.8 Reducing the system
?
x1+ 7x2
= y1
3x1+ 20x2
= y2
?
, we obtain
?x1
=
−20y1
+
7y2
x2
=
3y1
−
y2
?
.
2.1.9 We have to attempt to solve the equation
?y1
y2
?
=
?2
3
6
9
??x1
x2
?
for x1 and