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Unformatted text preview: 7.2 Matrix Representations and Similarity 1 Chapter 7. Change of Basis 7.2 Matrix Representations and Similarity Theorem 7.1. Similarity of Matrix Representations of T . Let T be a linear transformation of a Fnite-dimensional vector space V into itself, and let B and B be ordered bases of V . Let R B and R B be the matrix representations of T relative to B and B , respectively. Then R B = C 1 R B C where C = C B ,B is the change-of-coordinates matrix from B to B . Hence, R B and R B are similar matrices. Example. Page 406 number 2. Theorem. SigniFcance of the Similarity Relationship for Ma-trices. Two n n matrices are similar if and only if they are matrix represen-tations of the same linear transformation T relative to suitable ordered bases. 7.2 Matrix Representations and Similarity 2 Proof. Theorem 7.1 shows that matrix representations of the same trans-formation relative to diFerent bases are similar. Now for the converse. Let A be an n n matrix representing transformation...
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- Fall '04
- Vector Space