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Math Help - dimension

  1. #1
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    dimension

    Let V be a finite dimensional vector space. Show that if W1,....,Wn are subspaces of V such that none of these subspaces are qeual to V, then Union of all these subspaces does not equal V.

    i am thinking induction on dim V might help but do not really have idea how to go bout it. so please help.
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  2. #2
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    Quote Originally Posted by Kat-M View Post
    Let V be a finite dimensional vector space. Show that if W1,....,Wn are subspaces of V such that none of these subspaces are qeual to V, then Union of all these subspaces does not equal V.
    The union of two subpaces is a subspace if and only one is contained in another.
    Thus if W_1\cup W_2 is a subspace thus W_1\subseteq W_2 and W_2\subseteq W_1.
    And so \dim (W_1\cap W_2) = \max \{ \dim (W_1),\dim (W_2) \} < \dim (V).
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    Quote Originally Posted by ThePerfectHacker View Post
    The union of two subpaces is a subspace if and only one is contained in another.
    The reason that is so: Suppose u is in subspace U_1 but NOT subspace U_2 and v is in subspace U_2 but not in U_1. The u+ v cannot be in U_1\cup U_2. If it were, then it would have to be in either U_1 or U_2 (or both). If u+ v were in U_1, then, because U_1 is closed under addition and scalar multiplication u+ v+ (-u)= v would be in U_1, a cotradiction. If u+ v were in U_2, simlarly u+ v+ (-v)= u would be in U_2, again a contradiction.
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