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Unformatted text preview: Lecture 21 Let V, W be vector spaces, and let A : V W be a linear map. We defined the pullback operation A : W V . Last time we defined another pullback operator having the form A : k ( W ) k ( V ). This new pullback operator has the following properties: 1. A is linear. 2. If i k 1 ( W ) , i = 1 , 2, then A 1 2 = A 1 2 . 3. If is decomposable, that is if = 1 k where i W , then A = A 1 A k . 4. Suppose that U is a vector space and that B : W U is a linear map. Then, for every k ( U ), A B = ( BA ) . 4.7 Determinant Today we focus on the pullback operation in the special case where dim V = n . So, we are studying n ( V ), which is called the n th exterior power of V . Note that dim n ( V ) = 1. Given a linear map A : V V , what is the pullback operator A : n ( V ) n ( V )? (4.129) Since it is a linear map from a one dimensional vector space to a one dimensional vector space, the pullback operator A is simply multiplication by some constant A . That is, for all n ( V ), A = A . Definition 4.32. The determinant of A is det( A ) = A . (4.130) The determinant has the following properties: 1. If A = I is the identity map, then det( A ) = det( I ) = 1....
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This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.
- Fall '04
- Vector Space