# Composition of Linear Transformation and Matrix Multiplication

1. Mar 30, 2009

### jeff1evesque

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

2. Mar 30, 2009

### slider142

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.

3. Mar 31, 2009

### jeff1evesque

thanks.

Last edited: Mar 31, 2009