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...

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