Science:Math Exam Resources/Courses/MATH221/December 2008/Question 09 (b)
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Question 09 (b) 

Let and be a basis for (you do not have to prove this). Let be the linear transformation such that and . Find the matrix of T relative to the standard basis of . 
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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Science:Math Exam Resources/Courses/MATH221/December 2008/Question 09 (b)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let denote the standard basis for . We determined in part (a) that the matrix of T relative to the basis was , and we want to determine , the matrix of T relative to the basis . We can begin by writing
Let The above equality gives us a relation (check this!) for the coordinates of a vector with respect to each basis (recall that if , where , then the coordinates of with respect to the basis B is ). Now, we have by definition, and . Therefore, we can write
and so multiplying by , we find , from which we conclude that . To determine , we just need compute
and hence
