Let T be a linear operator on a finite-dimensional vector space V, and let B and B' be ordered bases for V. Suppose that Q is the change of coordinate matrix that changes B'- coordinates into B-coordinates. Then [T] B¹ = Q=¹[T] BQ. Proof. Let I be the identity transformation on V. Then T = IT = TI; hence, by Theorem 2.11 (p. 89), Q[T], [1] [T] = [IT]} = [TI] = [T] [I]} = [T],Q. = Therefore [T] =Q-¹[T] BQ. B¹

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