MATH221 December 2008
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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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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
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