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Math Help - Direct Sum of Dimensions

  1. #1
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    Direct Sum of Dimensions

    I have a question here on direct sums which states:

    Let U and V be subspaces of R5 such that U + V = R5, dimU = 2, and dimV = 3. Show that R5 = U is a direct sum of V.

    This is how I approached it:

    R5 = U is a direct sum of V
    dim(U+V)= 5 = dimR5

    If U is contained in V, dimU = dimV => U=V

    Therefore: U+V is in R5
    R5 = U is a direct sum of V.


    Does that make any sense? Could someone please review this and tell me if I am correct, or atleast steer me on the right path.
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  2. #2
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    \dim(U+V)=\dim(U)+\dim(V)-\dim(U\cap V), that leads \dim(U\cap V)=0, so U\oplus V=\mathbb R^5.
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  3. #3
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    Quote Originally Posted by GreenDay14 View Post
    I have a question here on direct sums which states:

    Let U and V be subspaces of R5 such that U + V = R5, dimU = 2, and dimV = 3. Show that R5 = U is a direct sum of V.

    This is how I approached it:

    R5 = U is a direct sum of V
    dim(U+V)= 5 = dimR5

    If U is contained in V, dimU = dimV => U=V

    Therefore: U+V is in R5
    R5 = U is a direct sum of V.


    Does that make any sense? Could someone please review this and tell me if I am correct, or atleast steer me on the right path.
    I'm not really sure what you tried to do there, but you only need to show that U \cap V= \{ 0 \}
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