Page 186 - A Course in Linear Algebra with Applications

170 Chapter Six: Linear Transformationns

Therefore we obtain the fundamental relation

M B = S[v] B>.

Thus left multiplication by the change of basis matrix S trans-
forms coordinate vectors with respect to B' into coordinate
vectors with respect to B. It is in this sense that the matrix
S describes the basis change B' —> B. Here it is important
to observe how S is formed: its ith column is the coordinate
vector of v[, the ith vector of B', with respect to the basis B.
It is a crucial remark that the change of basis matrix S
is always invertible. Indeed, if this were false, there would
by 2.3.5 be a non-zero n-column vector X such that SX — 0.
However, if u denotes the vector in V whose coordinate vector
with respect to basis B' is X, then [u]g = SX = 0, which can
only mean that u = 0 and X = 0, a contradiction.
As one would expect, the matrix S~ x represents the in-
verse change of basis B —> B'\ for the equation M s = ^ M s '
implies that
= S-\v\ B.
[v\ BI

These conclusions can be summed up in the following
form.
Theorem 6.2.4
Let B and B' be two ordered bases of an n-dimensional vector
space V. Define S to be the n x n matrix whose ith column is
the coordinate vector of the ith vector of B' with respect to the
basis B. Then S is invertible and, ifv is any vector ofV,

1
M s = S[v] B> and [v] B/ = S~ [v] B.