If you submitted the early homework by 5:00 Wednesday, you don't need to
turn in
1.8 #14 and 25, 1.9 #13 and 29, or 1.10 #5 on Friday.
Question 1:Consider the following matrices:
| and M= |
| 1 | 4 | -3 | -3 | -27 |
| 1 | 5 | -1 | -4 | -22 |
| -3 | 1 | 5 | -4 | 26 |
|
. |
Recall that we've worked with M before.
If you think they're relevant, you may cite earlier results about M
without deriving them again.
For which of these matrices are the associated linear transformations
one-to-one? For which are they onto? Justify your answers. (A
sentence or two should suffice.)
Question 2:
Let T be the linear transform from R2 to
itself which first rotates the plane 90 degrees counterclockwise, then
reflects it across the line y=x, and finally projects
vertically onto the x-axis.
Find the matrix associated to T.
(Note: I can think of at least three good ways to approach questions
like this. My experience is that it's generally easier to work out
all the T(ei) to get the columns of the matrix.)
Question 3: Let L
be a linear transformation from R3
to R, with
L(e1+e2+e3)=5
and
L(e1-e2+e3)=1.
Find L(e1+e3).