assign4-sol

assign4-sol - Math 235 Assignment 4 Due 9:15am, Wednesday...

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Unformatted text preview: 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 different 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 ) = { } . 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 ) , and we indicate this by writing V = ker P ker( I- P ) . A familiar example is R 2 = R R . ) 1 Rubriccomes from the Latin word for red. Printers used to print important instructions in red ink so that readers would not miss them. Alas, this is no longer done. There was a universal but unwritten rubric, namely, Always read the rubric. The word appeared in Mediaeval medicine and up to quite recently, for the signs for inflammation were listed as rubor (redness), calor (heat), tumor (swelling) and dolor (pain). It was a catchy mnemonic, at least for people in earlier times when Latin was the language of mathematics, science and medicine. 1 vii. Prove that P is diagonalisable. viii. If dim( P V ) = k (the rank of P ) and dim V = n, find the characteristic polynomial of P and prove your assertion . (b) Let L be the line lx + my = 0 in R 2 that passes through the origin (0 , 0) of the coordinate system. Let the point ( x 1 , y 1 ) be the foot of the perpendicular drawn from a point ( x , y ) to L. Let T : R R : ( x , y ) ( x 1 , y 1 ) . i. Prove that the line passing through ( x , y ) and perpendicu- lar to the line L has the equation- mx + ly =- mx + ly ....
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assign4-sol - Math 235 Assignment 4 Due 9:15am, Wednesday...

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