Composition of Linear Transformation and Matrix Multiplication

Theorem 2.15:
Let A be an m x n matrix with entries from F. Then the left-multiplication transformation
[tex]L_A: F^n --> F^m[/tex]. Furthermore, if B is any other m x n matrix ( with entries from F ) and B and D are the standard ordered bases for [tex]F^n and F^m[/tex], respectively, then we have the following properties.

(d.) If [tex]T: F^n --> F^m[/tex] is linear, then there exists a unique m x n matrix C such that [tex]T = L_C[/tex]. In fact [tex]C = [T]_B ^D[/tex]

proof: Let [tex] C = [T]_B ^D[/tex]. By Theorem 2.14, we have [tex][T(x)]_D = [T]_B ^D[x]_B[/tex] or T(x) = Cx = [tex]L_C(x)[/tex] for all x in [tex]F^n[/tex]. So T = [tex]L_C[/tex]

In particular I dont understand how T(x) = Cx


Thanks,


JL
 
1,005
64
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 [itex][T]_B ^D[x]_B[/itex], the next line is just replacing symbols with their equivalent matrix/vector forms.
 
thanks.
 
Last edited:

Related Threads for: Composition of Linear Transformation and Matrix Multiplication

Replies
4
Views
3K
Replies
1
Views
4K
Replies
7
Views
1K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top