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Question 09 (a)
Let be a basis for and let be the linear transformation such that and . Find the matrix of T relative to the basis B.
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In general, if is a basis for , and T is a linear transformation mapping , we know that for each j, is a linear combination of the basis vectors, i.e.
Recall T is completely determined by its action on the basis vectors. So the coordinates completely characterize the matrix T, in the sense that if we know all the , then we know exactly the linear transformation T. Therefore, the matrix whose entry in the ith row and jth column is is called the matrix of T relative to the basis .
For this problem, we are given that and . Therefore, writing
we identify . Hence the matrix of T relative to the given basis B is