Page 620 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

such that 1 holds for all vectors x in V, meaning that A times the coordinate vector of x equals the coordinate vector of the image
of . In particular, we want this equation to hold for the basis vectors , ,…, ; that is,
(2)

But

so

Substituting these results into 2 yields (3)
(4)
which shows that the successive columns of A are the coordinate vectors of
with respect to the basis . Thus the matrix for T with respect to the bases B and is
This matrix will be denoted by the symbol
so the preceding formula can also be written as
and from 1, this matrix has the property