problemset5 - MAS 213 Linear Algebra II Problem list for...

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MAS 213: Linear Algebra II. Problem list for Week #5. Tutorial during the lecture on 4th October. This week’s topics: The co-ordinate vector of a vector with respect to a basis. The matrix representation of a linear transformation with respect to choices of basis for the domain and co-domain. Tutorial problems: Problem 1: (Problem 2.2.2 (selected parts) from [FIS].) Here we will consider several linear transformations T : R n R m . We let β denote the standard ordered basis of the domain, and γ denote the standard ordered basis of the co-domain. For each linear transformation below, determine [ T ] γ β , the matrix representation of the given T with respect to these standard ordered bases. (i) T : R 3 R 3 defined by T ( a 1 , a 2 , a 3 ) = (2 a 2 + a 3 , - a 1 + 4 a 2 + 5 a 3 , a 1 + a 3 ) . (ii) T : R n R n defined by T ( a 1 , . . . , a n ) = ( a 1 , . . . , a 1 ) . (iii) T : R n R n defined by T ( a 1 , . . . , a n ) = ( a n , a n 1 , . . . , a 2 , a 1 ) . (iv) T : R n R defined by T ( a 1 , . . . , a n ) = a 1 + a n . 1
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Problem 2: (Problem 2.2.3 from [FIS].) Consider the linear transformation T : R 2 R 3 defined by the formula T ( a 1 , a 2 ) = ( a 1 - a 2 , a 1 , 2 a 1 + a 2 ) .
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