200 CHAPTER 9. LINEAR TRANSFORMATIONS
Describe how to consider both linear transformations and translations all at once byforming appropriate 4×4 matrices.
30. You want to add(
1, 2, 3)
to every point in R3 and then rotate about the zaxis counter clockwise through an angle of 30◦. Find what happens to the point(
1, 1, 1).
31. Write the solution set of the following system as the span of vectors and find a basisfor the solution space of the following system. 1 −1 2
1 −2 13 −4 5
x
yz
=
000
.
32. Using Problem 31 find the general solution to the following linear system. 1 −1 21 −2 13 −4 5
x
yz
=
124
.
33. Write the solution set of the following system as the span of vectors and find a basisfor the solution space of the following system. 0 −1 2
1 −2 11 −4 5
x
yz
=
000
.
34. Using Problem 33 find the general solution to the following linear system. 0 −1 21 −2 11 −4 5
x
yz
=
1−11
.
35. Write the solution set of the following system as the span of vectors and find a basisfor the solution space of the following system. 1 −1 2
1 −2 03 −4 4
x
yz
=
000
.
36. Using Problem 35 find the general solution to the following linear system. 1 −1 21 −2 03 −4 4
x
yz
=
124
.