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Unformatted text preview: Math 223, Section 102, Homework no. 6 (due Wednesday, October 27, 2010) [8 exercises, 35 points + 7 bonus points] Exercise 1 (6 points) . Consider the matrix A = 1 1 1 1 1 2 3 1 1 2 2 1 0 1 ∈ M 4 × 4 ( R ) . (1) (3 points) Is A invertible? If yes, compute its inverse. Now consider the linear transformation T : P 3 ( R ) → P 3 ( R ) , f ( x ) 7→ f ( x + 1) + x 2 f ( 1) + x 3 f (0) . We use the standard ordered basis β = (1 ,x,x 2 ,x 3 ) for P 3 ( R ). (2) (1 point) Determine the matrix [ T ] β β . (3) (2 points) Is T invertible? If yes, give a formula for T 1 ( f ) for arbitrary f ∈ P 3 ( R ). Exercise 2 (6 points + 2 bonus points) . Consider the linear transformation L A : R 2 → R 2 (see Assignment No. 5, Exercise 9) for the matrix A = 1 10 17 9 9 7 . (1) (1 point) Let β = ( e 1 ,e 2 ) be the standard ordered basis for R 2 . Determine [ L A ] β β . (2) (3 points) Compute the matrices [ I R 2 ] β γ and [ I R 2 ] γ β for the ordered basis γ = 1 √ 10 3 1 , 1 √ 10 1 3 ....
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This note was uploaded on 09/30/2011 for the course MATH 221 at UBC.
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