Direct sum problem

**tttcomrader** · April 16th 2008, 11:30 AM

Let $W_{1},W_{2},K_{1},K_{2},...,K_{p},M_{1},M_{2},..., M_{q}$ be subspaces of a vector space V such that $W_{1}=K_{1} \oplus K_{2} \oplus \cdot \cdot \cdot \oplus K_{p}$ and $W_{2} = M_{1} \oplus M_{2} \oplus \cdot \cdot \cdot \oplus M_{q}$

Prove that if $W_{1} \cap W_{2} = \{ 0 \}$ , then $W_{1}+W_{2} = W_{1} \oplus W_{2} = K_{1} \oplus K_{2} \oplus \cdot \cdot \cdot \oplus K_{p} \oplus M_{1} \oplus M_{2} \oplus \cdot \cdot \cdot M_{q}$

Proof:

Now, $W_{1} + W_{2} = W_{1} \oplus W_{2}$ is quite simple since their intersection is empty.

Now, since all the subspaces inside W1 and W2 doesn't intersect with one another by the defintion of direct sum, that means $W_{1} \oplus W_{2} = K_{1} \oplus K_{2} \oplus \cdot \cdot \cdot \oplus K_{p} \oplus M_{1} \oplus M_{2} \oplus \cdot \cdot \cdot M_{q}$

This looks too simple, is that right?

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