The first thing to do is to compute

, the matrix for the linear transformation

with respect to the basis

.

It will be a 3 by 3 matrix, as 3 is the dimension of our vector space. Each row and column will stand for a basis vector; lets name them

. The columns and rows of the matrix will correspond to our basis vectors; say the columns and rows correspond to

respectively. We need to do the following:

Find the image of each basis vector under

. Write these images out in terms of our basis. Then we have

; that is,

**the entry in column j, row i of is the coefficient of in the expression for **.

Let's try it for our first basis vector

:

Thus, the first column in the matrix should be