Brief overview of the development of the relationship between a linear transformation T and its associated matrix M.

Assumptions:
Vector spaces and bases have been discussed. In addition, the fact that a linear transformation is completely determined by its action on a basis has been covered.

Sketch of the development:
Let S = {e₁,e₂, ... , e_n} be the standard (natural) basis for Rⁿ. Then for v in Rⁿ we can express v as a linear combination of the vectors in S:

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It follows by the linearity property for linear transformations that

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Hence the m by n matrix M associated with the linear transformation T is given by

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Since T(e_j) is in R^m, matrix M is m by n.