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# ass4 - Math 235 Assignment 4 Due 9:15am Wednesday 1 From...

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Math 235 Assignment 4 Due 9:15am, Wednesday Feb 14, 2007. 1. From the Text § 5.3 #4, #18, and #26. § 5.4 #16 and #20. 2. Rubric: 1 In the following questions, the parts should be done in the order in which they are listed. (a) Let P : V V be a linear operator di ff erent from I , the identity operator in V , and from O , the zero operator on V , and such that P 2 = P ( P is said to be an idempotent operator). i. Prove that I - P is an idempotent operator on V . ii. Recall that the range space of P is P V := { P u : u V} . Prove that A. ker( P ) = ( I - P ) V , B. ker( I - P ) = P V . ( Hint: Note that P ( I - P ) = O and that I = ( I - P ) + P . The latter is called a partition of unity .) iii. Prove that ker P ker( I - P ) = { 0 } . iv. Find the set of distinct eigenvalues of P . (This is called the spectrum of P , and is denoted by spec ( P ) . ) v. Prove that P is not invertible. vi. Prove that if v V then there exists a unique pair ( v 1 , v 2 ) of vectors with v 1 ker P and v 2 ker( I - P ) such that v = v 1 + v 2 . ( Comment: In this case we say that V is a direct sum of the subspaces ker P and ker( I - P ) ,

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ass4 - Math 235 Assignment 4 Due 9:15am Wednesday 1 From...

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