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081st_tut8sol - MATH1111/2008-09/Tutorial VIII Solution 1...

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MATH1111/2008-09/Tutorial VIII Solution 1 Tutorial VIII Suggested Solution 1. Is it possible to construct a linear transformation fulfilling the following requirements? Describe the linear transformation if yes, or give an explanation if no. (a) T : R 2 R 2 , T ((1 0) T ) = (2 0) T , T ((0 1) T ) = (0 2) T , T ((1 1) T ) = (2 - 2) T . (b) T : R 2 R 2 , T ((1 - 1) T ) = (2 0) T , T ((2 - 1) T ) = (0 2) T , T (( - 3 2) T ) = ( - 2 - 2) T . (c) T : R 3 R 2 , T ((1 - 1 1) T ) = (2 0) T , T ((1 1 1) T ) = (0 2) T . Find their standard matrix representations for the cases of linear transformations. Ans . (a) No. It is because (1 1) T = (1 0) T + (0 1) T . If such a linear transformation exists, then T ((1 1) T ) = T ((1 0) T + (0 1) T ) = T ((1 0) T ) + T ((0 1) T ) = (2 0) T + (0 2) T ) = (2 2) T . i.e. We cannot assign T ((1 1) T ) = (2 - 2) T . (b) Yes. Define T : R 2 R 2 by T ( α (1 - 1) T + β (2 - 1) T ) = α (2 0) T + β (0 2) T . Then T (( - 3 2) T ) = ( - 2 - 2) T . In terms of standard coordinate vectors, T (( x y ) T ) = ( - 2 x - 4 y 2 x + 2 y ) T . The standard matrix representation is - 2 - 4 2 2 . (c) Yes. To construct it , we pick a vector u in R 3 so that (1 - 1 1) T , (1 1 1) T , u constitute a basis. To fulfill the requirement, we can assignment T ( u ) to any vector in R 2 . e.g. We may take u = (0 0 1) T , and set T ((0 0 1) T ) = 0 . An example is the linear transformation T : R 3 R 2 , T ( α (1 - 1 1) T + β (1 1 1) T + γ (0 0 1) T ) = α (2 0) T + β (0 2) T .
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