Page 621 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(4a)

Remark Observe that in the notation the right subscript is a basis for the domain of T, and the left subscript is a basis

for the image space of T (Figure 8.4.3). Moreover, observe howthe subscript B seems to “cancel out” in Formula 4a (Figure

8.4.4).

Figure 8.4.3

Figure 8.4.4

Matrices of Linear Operators

In the special case where (so that is a linear operator), it is usual to take when constructing a matrix

for T. In this case the resulting matrix is called the matrix for T with respect to the basis B and is usually denoted by

rather than . If , then Formulas 4 and 4a become

(5)
and

Phrased informally, 4a and 5a state that the matrix for T times the coordinate vector for is the coordinate vector for (5a)
.

EXAMPLE 1 Matrix for a Linear Transformation
Let be the linear transformation defined by
Find the matrix for T with respect to the standard bases
where

Solution

From the given formula for T we obtain