problemset6solutions - MAS 213 Linear Algebra II Problem...

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Unformatted text preview: MAS 213: Linear Algebra II. Problem list for Week #6. Tutorial on 12th October. Problem 1: (Problems 2.3.3 in [FIS].) Consider two linear transformations: • T : P 2 ( R ) → P 2 ( R ) defined by T ( p ( x )) = (3 + x ) p ′ ( x ) + 2 p ( x ). • U : P 2 ( R ) → R 3 defined by U ( a + bx + cx 2 ) = ( a + b,c,a − b ). Let β and γ be the standard ordered bases of P 2 ( R ) and R 3 . (1) Check that the expected equation [ U ◦ T ] γ β = [ U ] γ β [ T ] β holds. (2) Let h ( x ) = 3 − 2 x + x 2 ∈ P 2 ( R ). Check that the expected equation [ U ( h ( x ))] γ = [ U ] γ β [ h ( x )] β holds. Solution. We need to calculate various ingredients for this problem. First, to determine [ T ] β : • T (1) = 2 = (2) ∗ 1 + (0) ∗ x + (0) ∗ x 2 . • T ( x ) = 3 + 3 x = (3) ∗ 1 + (3) ∗ x. • T ( x 2 ) = 4 x 2 + 6 x = (0) ∗ 1 + (6) ∗ x + (4) ∗ x 2 . [ T ] β = [ T ] β β =   2 3 0 0 3 6 0 0 4   . Next, to determine [ U ] γ β : • U (1) = (1 , , 1) = (1) ∗ (1 , , 0) + (0) ∗ (0 , 1 , 0) + (1) ∗ (0 , , 1) . • U ( x ) = (1 , , − 1) = (1) ∗ (1 , , 0) + (0) ∗ (0 , 1 , 0) + ( − 1) ∗ (0 , , 1) . • U ( x 2 ) = (0 , 1 , 0) = (0) ∗ (1 , , 0) + (1) ∗ (0 , 1 , 0) + (0) ∗ (0 , , 1) . Thus: [ U ] γ β =   1 1 0 0 1 1 − 1 0   . Finally, to calculate [ U ◦ T ] γ β : • ( U ◦ T )(1) = U (2) = (2 , , 2) = (2) ∗ (1 , , 0) + (0) ∗ (0 , 1 , 0) + (2) ∗ (0 , , 1) . • ( U ◦ T )( x ) = U (3 + 3 x ) = (6 , , 0) = (6) ∗ (1 , , 0) + (0) ∗ (0 , 1 , 0) + (0) ∗ (0 , , 1) . • ( U ◦ T )( x 2 ) = U (6 x +4 x 2 ) = (6 , 4 , − 6) = (6) ∗ (1 , , 0)+(4) ∗ (0 , 1 , 0)+ ( − 6) ∗ (0 , , 1) . Thus: [ U ◦ T ] γ β =   2 6 6 0 0 4 2 0 − 6   . 1 2 So, we can finally check that the required equation does indeed hold: [ U ◦ T ] γ β =   2 6 6 0 0 4 2 0 − 6   =   1 1 0 0 1 1 − 1 0   ·   2 3 0 0 3 6 0 0 4   = [ U ] γ β [ T ] β . We move on to part 2 of this problem. There are two extra pieces we need, namely: [ h ( x )] β =   3 − 2 1   and [ U ( h ( x ))] γ = [(1 , 1 , 5)] γ =   1 1 5   . Now we can check the required equation: [ U ( h ( x ))] γ =   1 1 5   =   1 1 0 0 1 1 − 1 0     3 − 2 1   = [ U ] γ β [ h ( x )] β . Tick. 3 Problem 2: (Problems 2.4.19 in [FIS].) Let T : M 2 × 2 ( F ) → M 2 × 2 ( F ) be the linear transformation which sends a matrix to its transpose. Let β be the ordered basis β = { E 11 ,E 12 ,E 21 ,E 22 } for M 2 × 2 ( F ) (where E ij is the 2 × 2 matrix which has a zero in every position except for a single 1 in the row i column j entry). Let A denote the matrix representation [ T ] β . Let M = [ 1 2 3 4 ] ∈ M 2 × 2 ( F )....
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This note was uploaded on 12/08/2010 for the course SPMS MAS213 taught by Professor Andrewkricker during the Fall '10 term at Nanyang Technological University.

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problemset6solutions - MAS 213 Linear Algebra II Problem...

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