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Math Help - Direct Sum Question

  1. #1
    Junior Member
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    Direct Sum Question

    Hi everyone,

    I am having a bit of difficulty finding an example that works for this question. It asks to find subspaces W1, W2, and K, in vector space V such that

    V = K *direct sum* W1
    V = K *direct sum* W2

    but W1 does not equal W2.

    I have considered using K as R^n, and W1 as 0, and W2, as one vector in R^n. however, K and W2 have a non-zero intersection. Am I thinking in the wrong vector space?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Direct Sum Question

    Choose for example the subspaces of V=\mathbb{R}^2:

    W_1=\{(\lambda,0):\lambda\in \mathbb{R}\},\;W_2=\{(0,\mu):\mu\in \mathbb{R}\},\;K=\{(\gamma,\gamma):\gamma\in \mathbb{R}\}
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