10aAss4 - AS4/MATH1111/YKL/10-11 THE UNIVERSITY OF HONG...

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AS4/MATH1111/YKL/10-11 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1111: Linear Algebra Assignment 4 Due date : Nov 25, 2010 before 6:30 p.m. Part I: Hand-in your solutions 1. Let a = ± 1 3 ² ; b = ± 2 2 ² ; c = ± 3 1 ² be three vectors in R 2 . (a) Is it possible to ±nd a linear transformation L : R 2 ! R 2 such that L ( a ) = b ; L ( b ) = c ; L ( c ) = a ? Explain your answer. (b) De±ne the linear transformation T : R 2 ! R 2 by T ( ± x y ² ) = ± ± 8 x + 6 y ± 9 x + 7 y ² : i. Evaluate the matrix representation A of T relative to the ordered bases E = [ a ;b ] and F = [ b ;c ], i.e. [ T ( x )] F = A [ x ] E for all x 2 R 2 . ii. Find an ordered basis G of R 2 such that the matrix representation of T with respect to G (i.e. relative to G and G ) is a diagonal matrix. 2. (a) Let A = 0 @ 1 1 2 0 1 1 1 3 4 1 A . i. Show that R 3 = N ( A ) ² c ( A ), where N ( A ) is the nullspace and c ( A ) is the column space of A . ii. Is it always true that R 3 = N ( B ) + c ( B ) for any 3 ³ 3 matrix B ? (b) Let L : V ! V be a linear operator. If L ´ L = L , i.e. L ( v ) = L ( L ( v )), show that V = ker( L ) ² L ( V ).
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This note was uploaded on 05/04/2011 for the course MATH 1111 taught by Professor Dr,li during the Spring '10 term at HKU.

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10aAss4 - AS4/MATH1111/YKL/10-11 THE UNIVERSITY OF HONG...

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