Chapter 3 Linear transformations

Solution: By the above resultP=[[1] B[1+x] B[1+x+x 2 ] B].[1] B=(1,0,0),[1+x] B=(1,1/2,0),[1+x+x 2 ] B]=(1,1/2,1/3).Hence P=⎡⎢⎢⎢⎣10011/ 201⎤⎥⎥⎥⎦1/ 21/ 3.Theorem 3.3.5Let T:V → V be a linear operator on a finite dimensional vector space V, and let B and B'be bases for V.Then[T] B=P − 1where P is the transition matrix from B' to B.[T] B'P,Definition 3.3.6Two n× n matrices A and B are said to be similar to each other if there is an invertiblematrix P such thatA=P −1AP.