Transition and coordinate matrices

[fattycakez]

Homework Statement

Consider the bases B = {b₁,b₂} and B' = {b'₁,b'₂} for R², where

b₁=(1, -1), b₂=(2,0), and b'₁=(1,2), b'₂=(1,-3)

a. Find the transition matrix P from B to B'
b. Compute the coordinate matrix [p]_B, where p=(4,3); then use the transition matrix P to compute [p]_B'

Homework Equations

The Attempt at a Solution

a. I found the transition matrix P to be
2/5 6/5
3/5 4/5

b. I found [p]_B to be (-3, 7/2)
and then I found [p]_B' to be (3,1)

Does this look correct? I am tripping out because this seems like something we covered in class months ago but this problem was just assigned this week. Am I missing something? Would the problem be different if B' represented an orthogonal basis? Any help is appreciated :)

 

[PeroK]

Does this look correct? I am tripping out because this seems like something we covered in class months ago but this problem was just assigned this week. Am I missing something? Would the problem be different if B' represented an orthogonal basis? Any help is appreciated :)

You should be able to check your answers for yourself on this one. For example, to check the vectors are correct you simply check that:

$-3b_1 + (7/2)b_2 = -3(1, -1) + 7/2(2, 0) = (-3, 3) + (7, 0) = (4,3)$

So, (4, 3) is indeed (3, 7/2) in basis B.

Etc.

And, you can use the transition matrix to operate on the vectors to check that it does map the vectors correctly. This is often a good thing to check in any case.

Everything looks correct, but I'd suggest you check them yourself.