6.47(a), §2 Solution

The basis vectors of the S-system are given in terms of the E-system, so, to get the transformation matrix, we simply put them as columns of the matrix:

Let's check this, just to be sure. The arbitrary vector can be expressed as

so

Since

the transformation matrix Q from S to E is

So

So a'=-5a+3b and b'=2a-b.