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∈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##

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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