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Unformatted text preview: Nullspaces of commuting transformations : LinearAlg Jonathan L.F. King University of Florida, Gainesville FL 326112082, USA squash@ufl.edu Webpage http://www.math.ufl.edu/ squash/ 26 October, 2011 (at 10:25 ) Abstract: Conditions under which the nullspace of a composition B A is the span of the two nullspaces. This is then applied to constantcoefficient linear differential equations. A Entrance The setting is a vectorspace V and two linear transformations A , B : V . Use both Nul( A ) and A for the nullspace of trn A . Use C for the composition C := BA . 1 : Fact. Suppose B , A : V V . Then Dim( B )+Dim( A ) 1 : > Dim Nul( BA ) 2 : > Dim( A ) . Now suppose that A B ( the trns commute ) so that C = BA = AB . Then, automatically, C Spn( B , A ) . 3 : We explore when we have equality in (1 ), and when in (3). For two subspaces W , W V , let W W mean that these { W , W } is a ( linearly ) independent set , as in ( * ), below. [ In particular, W and W only intersect in the singleton { } . ] More generally, K j =1 W j indicates mutual independence of the subspaces in that the only solution to w 1 + + w K = with each w j W j , * : is to have every w j be ....
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This note was uploaded on 01/26/2012 for the course MAC 3472 taught by Professor Jury during the Fall '07 term at University of Florida.
 Fall '07
 JURY
 Calculus, Transformations

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