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

 .