# Composition of Linear Transformation and Matrix Multiplication

by jeff1evesque
Tags: composition, linear, matrix, multiplication, transformation
 P: 312 Theorem 2.15: Let A be an m x n matrix with entries from F. Then the left-multiplication transformation $$L_A: F^n --> F^m$$. Furthermore, if B is any other m x n matrix ( with entries from F ) and B and D are the standard ordered bases for $$F^n and F^m$$, respectively, then we have the following properties. (d.) If $$T: F^n --> F^m$$ is linear, then there exists a unique m x n matrix C such that $$T = L_C$$. In fact $$C = [T]_B ^D$$ proof: Let $$C = [T]_B ^D$$. By Theorem 2.14, we have $$[T(x)]_D = [T]_B ^D[x]_B$$ or T(x) = Cx = $$L_C(x)$$ for all x in $$F^n$$. So T = $$L_C$$ In particular I dont understand how T(x) = Cx Thanks, JL
 P: 876 You previously defined C to be the matrix of T with respect to the bases B and D. By theorem 2.14, you have the equivalence to $[T]_B ^D[x]_B$, the next line is just replacing symbols with their equivalent matrix/vector forms.
 P: 312 thanks.

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