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