What is the Algebraic Proof of Linear Transformation Composition with Addition?

[autre]

Homework Statement

Prove:

Let V be a vector space over the field F . If A,B,C\in L(V) , then A\circ(B+C)=A\circ B+A\circ C .

The Attempt at a Solution

Note that A\circ B\in L(V) means A\circ B(\mathbf{v})=A(B(\mathbf{v})). Suppose (\alpha_{jk})_{j,k=1}^{n} and (\beta_{jk})_{j,k=1}^{n} are matrices of A and B and (\gamma_{jk})_{j,k=1}^{n} is a matrix of C . Then, B+C=(\beta_{jk}+\gamma_{jk})_{j,k=1}^{n} and A\circ(B+C)=A((B+C))=\sum_{i=1}^{n}\alpha_{ji}(\beta_{ik}+\gamma_{ik})...

I'm a little stuck at this point. Any ideas?

 

[Deveno]

you just need to continue the algebra a little further...

\sum_i \alpha_{ji}(\beta_{ik} + \gamma_{ik}) = \left(\sum_i\alpha_{ji}\beta_{ik}\right) + \left(\sum_i\alpha_{ji}\gamma_{ik}\right) = \dots