College Algebra Exam Review 171

College Algebra - 3.4 THE DUAL OF A VECTOR SPACE AND MATRICES Thus the j t h column of ŒS T D;B and of ŒS D;C ŒT C;B agree 181 I For a

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Unformatted text preview: 3.4. THE DUAL OF A VECTOR SPACE AND MATRICES Thus the j t h column of ŒS T D;B and of ŒS D;C ŒT C;B agree. 181 I For a vector space V over a field K , we denote the set of K –linear maps from K to K by EndK .V /. Since the composition of linear maps is linear, EndK .V / has a product .S; T / 7! S T . The reader can check that EndK .V / with the operations of addition and and composition of linear operators is a ring with identity. To simplify notation, we write ŒT B instead of ŒT B;B for the matrix of a linear transformation T with respect to a single basis B of V . Corollary 3.4.12. Let V be a finite dimensional vector space over K . Let n denote the dimension of V and let B be an ordered basis of V . (a) For all S; T 2 EndK .V /, ŒS T B;B D ŒS B ŒT B . (b) T 7! ŒT B is a ring isomorphism from EndK .V / to Matn .K/. Lemma 3.4.13. Let B D .v1 ; : : : ; vn / and C D .w1 ; : : : ; wn / be two bases of a vector space V over a field K . Denote the dual bases of V by B D v1 ; : : : ; vn and C D w1 ; : : : ; wn . Let id denote the identity linear transformation of V . (a) The matrix ŒidB;C of the identity transformation with respect to the bases C and B has .i; j / entry hwj ; vi i: (b) ŒidB;C is invertible with inverse ŒidC;B . Proof. Part (a) is immediate from the definition of the matrix of a linear transformation on page 179. For part (b), note that E D ŒidB D ŒidB;C ŒidC;B : I Let us consider the problem of determining the matrix of a linear transformation T with respect to two different bases of a vector space V . Let B and B 0 be two ordered bases of V . Then ŒT B D ŒidB;B 0 ŒT B 0 ŒidB 0 ;B ; by an application of Proposition 3.4.11. But the “change of basis matrices” ŒidB;B 0 and ŒidB 0 B are inverses, by Lemma 3.4.13 Writing Q D ŒidB;B 0 , we have ŒT B D QŒT B 0 Q 1 : ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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