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Math Help - Working with external and internal direct sums

  1. #1
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    Working with external and internal direct sums

    My information for this one is limited, so I'll need a hand.

    Suppose V and W are vector spaces over a field F.

    Now suppose W_1 and W_2 are subspaces of V. We denote the external direct sum of W_1 and W_2 by W_1\times W_2, and denote the internal direct sum of W_1 and W_2 by W_1\oplus W_2. Define \varphi : W_1\times W_2\rightarrow W_1+W_2 by \varphi(w_1,w_2)=w_1+w_2 for all (w_1,w_2)\in W_1\times W_2.

    a) Prove that \varphi is a surjective linear transformation.

    b) Prove that \varphi is an isomorphism if and only if W_1+W_2=W_1\oplus W_2.

    There isn't anything in our reading material that differentiates between external and internal direct sums, though I found the definitions online. In case anyone needs the definition for direct sum (in a basic sense), I have it below.

    V=W_1\oplus ...\oplus W_k, if W_1+...+W_k=V and W_1,...,W_k are independent.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    W_1+W_2 = \{w_1+w_2 : w_1 \in W_1, w_2 \in W_2\}. It's a subspace of V which contains both W_1 and W_2 (and in fact the smallest such subspace). In the event that every element of W_2+W_2 can be written uniquely as w_1+w_2, then we write W_2\oplus W_2 instead.

    Both are direct consequences of the definitions. It's very easy to see that \varphi is linear. To show that \varphi is surjective, take any w_1+w_2 \in W_1+W_2; then \varphi(w_1,w_2)=w_1+w_2.

    The second one isn't any harder! Give it a try and post your idea.
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