1. Suppose (a, a, , an) is a basis for Rn and let A = [a1 an]. Suppose bi, ,br R

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1. Suppose (a, a, , an) is a basis for Rn and let A = [a1 an]. Suppose bi, ,br R", and let (a) Find a linear map T : Rn Rm satisfying by expressing [T] in terms of A, B. Justify your answer. (Hint: What do A, B do to the standard basis vectors ei,ez,...,en?) (b) Show that there is a unique linear map T : Rn Rm satisfying (1). (Hint: Show that if U,T : Rn Rm are two such maps, then U = T.) This shows that a linear map is uniquely determined by where it sends a basis, and tells us how to find the matrix for that map.

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