Linear Algebra - Bilinear Forms and Change of Basis

TorcidaS

1. The problem statement, all variables and given/known data
Find the matrix of f relative to Alpha' and Beta'.

Alpha' = [(1,0,0), (1,1,0), (2,-1,1)]
Beta' = [(-13,9), (10,-7)]


The question originally reads that f is a bilinear form.

I've found a (correct according to answer key) matrix A that is

3 -4
4 -5
-1 2

from a given basis of
Alpha = [(1,0,0), (1,1,0), (1,1,1)] and
Beta = [(1,-1), (2,-1)]

2. Relevant equations

If Q is the matrix of transition from Alpha to basis Alpha' of U and P is the matrix of transition from Beta to basis Beta' of V, then QTAP = matrix of f relative to Alpha' and Beta'.
T meaning transpose.

3. The attempt at a solution
I've gotten oh so far with this question, but I'm stuck here in the final part. I'm confused how to attain said matrices Q and P from the relevant equations. If I'm not terrible mistaken, I think I have to do something involving inverses but I'm at a loss here.

Any help will of course be greatly appreciated.

vela

You want to express the vectors of one basis in terms of the other basis.