# Explanation of a Line of a proof in Axler Linear Algebra Done Right 3r

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• MidgetDwarf

#### MidgetDwarf

∈Was wondering if anyone here could help me with an explanation as to how Axler arrived at a particular step in a proof.

These are the relevant definitions listed in the book:

Definition of Matrix of a Linear Map, M(T):

Suppose ##T∈L(V,W)## and ##v_1,...,v_n## is a basis of V and ##w_1 ,...,w_m## is a basis of W. The matrix of T with respect to these bases is the m-by-n matrix M(T) whose entries ##A_j , _k## are defined by ## T_v_k = A_1,k w_1 + ... +A_m,k w_m ##

Definition of Matrix Multiplication:

Suppose A is an m-by-n matrix and C is an n-by-p matrix. Then AC is defined to the m - by- p matrix whose entry in row j, column k, is given by the following equation: ## (AC)_{j,k} = \sum_{r=1}^n A_j,r C_r,k ##

Now for the Theorem of the proof I need help with.

Theorem 3.43 (page 74-75): The Matrix Of The Product Of Linear Maps:

If T∈L(U,V) and SεL(V,W) , then M(ST)=M(S)M(T).

Proof:

Assume ## v_1 , ... , v_n ## is a basis of V and ##w_1 , ... , w_m ## is a basis of W. Suppose also that we have another vector space U and that ## u_1 ,..., u_p ## is a basis of U. Consider linear maps T : U →V and S : V→W. ( I proved easier that the composition of linear maps is a linear maps)

Suppose M(S) = A and M(T) + C. For 1≤ k ≤ p , we have

##(ST)u_k ## = ## S(\sum_{r=1}^n C_r,k v_r ) = \sum_{r=1}^n C_r,k Sv_r ## ##= \sum_{r=1}^n C_r,k \sum_{j=1}^m A_j,r w_j##

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Hi. There seem to be some formatting issues with the post. Please resolve those.

Wow my Latek came out wrong. Ill try some practice with easier code, before reposting this question. If anyone is interested, it is the 3rd line to 4th line on page 74 of Axler Linear Algebra Done Right 3rd ed.

Wow my Latek came out wrong. Ill try some practice with easier code, before reposting this question. If anyone is interested, it is the 3rd line to 4th line on page 74 of Axler Linear Algebra Done Right 3rd ed.
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