Question

Let β be the standard basis of R^n. Let T be a linear map whose coordinate matrix with regard to the standard basis is denoted by T. Let P and Q be two basis change matrices, where P and Q are two new bases of R^n. If Q = PTP-1, find the matrix T.

a) QP
b) PQ
c) P-1Q
d) TP-1Q

Answer 1

The matrix T in the equation Q = PTP-1 is T = Q(P-1).

Step-by-step explanation:

To find the matrix T in the given equation Q = PTP-1, we need to solve for T. We can start by multiplying both sides of the equation by P-1 on the right side:

Q(P-1) = (PT)(P-1)T

Simplifying, we get:

Q(P-1) = InT

Since P-1P = In, where In is the identity matrix of size n, we have:

Q(P-1) = IT = T

Therefore, the matrix T is simply the same as Q(P-1).