Linear Transformations and Coordinate

[Horland]

Homework Statement

Let B be a basis of R^2 consisting of the vectors <5,2> and <1,5> and let R be the basis consisting of <2,3> and <1,2>

Find a matrix P such that [x]_r=P[x]_b for all x in R^2

Homework Equations

Ax=B?

The Attempt at a Solution

I attempted by using Ax=B as a format to solve for P, or x in the equation. I took the inverse of the column matrix of B because B is a basis, and if I am correct, a matrix's basis is defined as its coordinates ([x]_b).

I understand the geometrical interpretation of a linear transformation, yet I have no idea how to represent the transformation as a matrix. I would love an in depth description to approach similar exercises.

 

[gabbagabbahey]

Homework Equations

Ax=B?

Not quite. The relevant equation in this case is the one given in the problem statement.

$$[x]_r=P[x]_b$$


How would you go about represented a general vector $x$ in the B-basis?