4.4.Change of basis: Linear transformations.

We mentioned above that a linear map is more fundamental than the matrix representing it relative to a chosen basis, for the matrix changes when we change the basis but the linear map remains unchanged. In this Section we will explore how the matrix of a linear map changes as we change the basis. We will first consider linear transformations (or endomorphisms), that is, linear maps .

As before we let be an -dimensional vector space with basis , and let be a linear transformation whose matrix relative to has entries defined by

Now let be another basis. Relative to this basis, will have a matrix with entries defined by

(15)

The obvious question is: How are the two matrices and related?

We can answer this question as follows. As we did in Section 2.3, we introduce the transition matrix relating the basis to the basis :

(16)

We can then compute both sides of equation (15) separately and compare. The right-hand side gives

On the other hand, the left-hand side gives

Comparing the two sides, we see that

(17)

or if we let , and denote the corresponding matrices

(18)

As we saw in Section 2.3, is invertible, with inverse whose entries are defined by

(19)

Let’s change the notation and denote by the matrix corresponding to . Multiplying both sides of equation (>18) by on the left, we obtain

(20)

or more explicitly

(21)