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# hw3 - A and consider the map L A B → A,B Show that L A is...

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Math 136A, Spring 2008 Problem Set #3 (due 4/18/2008) (1) Let V be the space of infinitely differentiable functions on the interval [0 , 1] and let p, q V . (a) Show that the map L : V V defined by L [ f ] = f 00 + pf 0 + qf is a linear map. (b) What can you say about the kernel of L ? (2) Let M ( n ) be the vector space of n × n matrices. The Lie bracket of two matrices A and B in M ( n ) is the matrix [ A, B ] = AB - BA (So [ A, B ] = 0 ⇐⇒ AB = BA .) Choose a matrix
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Unformatted text preview: A , and consider the map L A : B → [ A,B ] Show that L A is a linear map. Describe the kernel of L A . (3) Let V be the space of inﬁnitely diﬀerentaible functions on the interval [0 , 1]. Show that the map T : V → V deﬁned by ( Tf )( x ) = Z x f ( t ) dt is a linear map. Prove that T is injective. Is T surjective? (Why or whiy not?) 1...
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