Composition of Linear Transformation and Matrix Multiplication

[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 don't understand how T(x) = CxThanks,JL

 

[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.

 

[jeff1evesque]

thanks.