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nullspace

# nullspace - Nullspaces of commuting transformations...

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Nullspaces of commuting transformations : LinearAlg Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] 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 constant-coefficient 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 0 : > 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 0 ), and when in (3). For two subspaces W , W 0 V , let W W 0 mean that these { W , W 0 } is a ( linearly ) independent set , as in ( * ), below. [ In particular, W and W 0 only intersect in the singleton { 0 } . ] More generally, K j =1 W j indicates mutual independence of the subspaces in that the only solution to w 1 + · · · + w K = 0 with each w j W j , * : is to have every w j be 0 .

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