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Math Help - Direct sum problem

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    Direct sum problem

    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?
    Last edited by tttcomrader; April 16th 2008 at 02:48 PM.
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