converting a vector defined in terms of another basis

Question

This is for my computer graphics class and I'm trying to figure out the excersises. However I can't even quite understand the question

Given two non-standard orthonormal bases in 2D space, {a,b} and {c,d}, compute the 2*2 matrix that converts a vector defined in terms of {a,b} into that of {c,d}.

Does this mean to rotate {a,b} to {c,d}?

Answer 1

Let $ M = \begin{bmatrix} a_1 & b_1\\a_2 & b_2\end{bmatrix}. $

Let $ N = \begin{bmatrix} c_1 & d_1\\c_2 & d_2\end{bmatrix}. $

$T M = N \text{ iff } T = N M^t$

Calculating,

$ T = \begin{bmatrix} a_1 c_1 + b_1 d_1 & a_2 c_1 + b_2 d_1 \\ a_1 c_2 + b_1 d_2 & a_2 c_2 + b_2 d_2\end{bmatrix} $