Linear transformation algebra

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  • #1
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Homework Statement



Prove:

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

The Attempt at a Solution



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

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

Answers and Replies

  • #2
Deveno
Science Advisor
906
6
you just need to continue the algebra a little further....

[tex]\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[/tex]
 

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