HW 04
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moienHW 04
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moienMath 21b: Linear Algebra Spring 2016
Homework 4: Linear transformations
This homework is due on Monday, February 8, respectively on Tuesday February 9, 2016.
1 Which of the following transformations are linear? If it is, find
the matrix A whichimplements
the transformation.
7y x 2x + 22
x
a) T = −3x b) T y = y + 1
y
x z z − 2
x
2 x
x y + 2x − e
c) T = d) T y = z−x−y
y x
z
x 3y + x
e) T y = 0
z x
2 Find the inverse of the following linear transformations x → Ax
or state that it is not invertible
1 2 2 0
a) A = , b) A =
3 2 0 −3
0 1 3 −2
c) A = , d) A =
1 0 −6 4
d −b a b
e) Verify that /(ad − bc) is the inverse of A =
−c a c d
if ad 6= bc.
We learn how to invert a matrix later. For now, get the inverse
by solving Ax = ek , rendering the k’th column of A−1.
3 For each of the matrices, sketch the effect of the
linear transformation T (x) = Ax on the face.
0 1 0 −1 2 0 1 0 1 0
a) . b)
. c) . d)
. e)
.
1 0 1 0 0 2 0 −1 0 2
v1
4 a) Let v = v2 . Which matrix A implements the transformation
v3
x1
v2 x3 − v3 x2
x = x2 → v × x = v3x1 − v1x3 .
x3 v1x2 − v2x1
b) Which matrix A implements the transformation
x1
x = x2 → v · x = [v1x1 + v2x2 + v3x3] ?
x3
5 Find the linear transformation which rotates space counterclock-
wise around the z axes by an angle
π/4,
then reflects
at thexy-
1 0 0
plane. Draw the images of e1 = 0 , e2 = 1 and e3 = 0
0 0 1
and using these vectors to build the matrix.
Main properties
If A is a matrix then the map x → Ax = b is called a
linear transformation. It is invertible if x can be
obtained uniquely from b.
The columns of the matrix play a key role. The image
of the vector e1 is the first column, the image of e2 the
second column etc.