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problemset6_1 - MAS 213 Linear Algebra II Problem list for...

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MAS 213: Linear Algebra II. Problem list for Week #6. Tutorial on 12th October. This week’s topics: Matrix representations and compositions of linear transformations. Matrix representations and co-ordinate vectors. Invertible linear transformations and isomorphisms. Tutorial problems: 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. 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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