Math 2270 Spring 2004
Homework 4 Solutions
Section 2.1) 1, 2, 3, 8, 10, 14a, 16, 20, 22, 25, 26, 32, 42
(1) The transformation is not linear as the equation for y2 involves adding a constant.
(2) The transformation is linear.
(3) The transformation is not linear as the equation for y2 involves the product of x1 and
x3 . This product cannot be written in the matrix form required for linear transformations.
(8)
x
x + 7x =
x + 7x = y = −20y1 + 7y2 y1
1
1
1
2
2
1 →
→
0 + −x2 = y2 − 3y1 3x1 + 20x2 = y2 x2 = 3y1 − y2 (10)
(14a)
x1 + 2x2 = y1
4x1 + 9x2 = y2
"
2
5
→
3
k
#
x1 + 2x2 =
y1
0 + x2 = y2 − 4y1
→
"
1
5
1.5
k
#
→
"
→
1
0
x1
= 9y1 − 2y2
x2 = −4y1 + y2
1.5
k − 7.5
#
The matrix has rank equal to 2 when k 6= 7.5. Therefore, the matrix is invertible for all
values of k except 7.5.
(16, 25) T (~x) = 2~x for all ~x ∈ R2 . Geometrically, the transformation represents a dilation
by a factor of 2.
2x
= y1 x1
= y1 /2 1
→
2x2 = y2 x2 = y2 /2 The inverse transformation represents a dilation by a factor of 1/2 (a contraction).
(20, 26) T (e~1 ) = e~2 and T (e~2 ) = e~1 . For a general vector ~x, the x1 and x2 components are
swapped. Geometrically, the transformation reflects a vector over the line x1 = x2 .
x2 = y1
x1
= y2
→
x1
= y2
x2 = y1
The inverse transformation is the same as the original transformation. (It is a reflection in
the line x1 = x2 .)
1
Math 2270 Spring 2004
Homework 4 Solutions
(22) T (e~1 ) = e~1 and T (e~2 ) = −~
e2 . For a general vector ~x, the x1 component stays the same
and the x2 component becomes negative. Geometrically, the transformation reflects a vector
over the x1 − axis.
x
= y1 = y1 x1
1
→
x2 = −y2 −x2 = y2 The transformation is its own inverse. (The inverse transformation is also a reflection in the
x1 − axis.)
(32)
3I =
(42)


0


T  0  =
0
=


0


T  0  =
1
=


"
1


T  0  =
1
"
"
0
0
#
0
1
#
−1/2
1/2








3
0 ··· ···
0
3
0 ···
0
0
3
0
··· ··· ··· ···
0 ··· ··· ···
···
···
···
···
···
··· 0
··· 0
··· 0
··· ···
0
3

"
#
1
−1/2


T  0  =
−1/2
0





#
"
1
1/2


T  1  =
−1/2
0
#
#
"
1
1/2


T  1  =
1/2
1
Here, s is an arbitrary real number.
2









#
"
0
1


T  1  =
0
0





#
"
0
1


T  1  =
1
1
#
"
2s
0


T  s  =
0
s