NEED SOME HELP PLZ Let B be the basis of R2 consisting of th

NEED SOME HELP PLZ.

Let B be the basis of R^2 consisting of the vectors [5 -2] and [1 5], and let R be the basis consisting of [-2 -1] and [-3 -2]. Find a matrix P such that [x]_R = P[x]_B for all X in R^2. P = [ ]

Solution

Let R and B represent the matrices

[-2 -1]
[-3 -2]

and

[5 -2]
[1 5]

respectively (as given in the problem).

Then for any column vector x in R^2,
[x]r = R^(-1) [x] = R^(-1) (B[x]b) = (R^(-1) B)[x]b.

So if we choose P = R^(-1) B, then [x]r = P[x]b.

P = R^(-1) B

=
[-2/detR 1/detR][5 -2]
[3/detR -2/detR][1 5]

=
[-2/(4-3) 1/(4-3)][5 -2]
[3/(4-3) -2/(4-3)][1 5]

=
[-2 1][5 -2]
[3 -2][1 5]

=
[-9 9]
[13 -16].