Page 515 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition
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If we multiply the transition matrix from to B obtained in Example 1 and the transition matrix from B to obtained in
Example 2, we find
which shows that . The following theorem shows that this is not accidental.
THEOREM 6.5.1
If P is the transition matrix from a basis to a basis B for a finite-dimensional vector space V, then P is invertible, and
is the transition matrix from B to .
Proof Let Q be the transition matrix from B to . We shall show that and thus conclude that to complete the
proof.
Assume that and suppose that
From 5,
for all x in V. Multiplying the second equation through on the left by P and substituting the first gives
(7)
for all x in V. Letting in 7 gives
Similarly, successively substituting , …, in 7 yields
Therefore, .
To summarize, if P is the transition matrix from a basis to a basis B, then for every vector v, the following relationships hold:
(8)
