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Math Help - [SOLVED] Dimension

  1. #1
    Senior Member Spec's Avatar
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    [SOLVED] Dimension

    U_1 \cap U_2 = \{0\}, U_1 \subset V, U_2 \subset V

    Show that dim(U_1)+dim(U_2) \leq dim(V)
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  2. #2
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    Quote Originally Posted by Spec View Post
    U_1 \cap U_2 = \{0\}, U_1 \subset V, U_2 \subset V

    Show that dim(U_1)+dim(U_2) \leq dim(V)
    Let W be the sum of vector subspaces U_1 and U_2.
    Then W\subseteq V and so \text{dim} W \leq \text{dim} V.
    However, \text{dim} U_1 + \text{dim} U_2 = \text{dim} W.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    Let W be the sum of vector subspaces U_1 and U_2.
    Then W\subseteq V and so \text{dim} W \leq \text{dim} V.
    However, \text{dim} U_1 + \text{dim} U_2 = \text{dim} W.
    (1) dim(U_1\oplus U_2) = dim U_1+dim U_2
    (2) U_1\oplus U_2 \subseteq V

    (1), (2) \implies \text{dim} U_1 + \text{dim} U_2 \leq \text{dim} V

    Is that proof enough? I can prove (1), but not (2).
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  4. #4
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    Quote Originally Posted by Spec View Post
    Is that proof enough? I can prove (1), but not (2).
    Remember what the the sum of subspaces are. It is the smallest subspace that contains both U_1 and U_2.
    However, V already contains U_1,U_2, so W\subseteq V.
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