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Figure 4.3.4
Because of its importance, we shall state 5 as a theorem for future reference.
THEOREM 4.3.3
If is a linear transformation, and , , …, are the standard basis vectors for , then the standard matrix
for T is (6)
Formula 6 is a powerful tool for finding standard matrices and analyzing the geometric effect of a linear transformation. For
example, suppose that is the orthogonal projection on the -plane. Referring to Figure 4.3.4, it is evident
geometrically that
so by 6,
which agrees with the result in Table 5 of Section 4.2.
Using 6 another way, suppose that is multiplication by
The images of the standard basis vectors can be read directly from the columns of the matrix A:
