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Thread: Subspaces take 2.

  1. #1
    Super Member Showcase_22's Avatar
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    Subspaces take 2.

    Let $\displaystyle V=\mathbb{R}^4$. Suppose further that $\displaystyle W_1$ is a subsapce of $\displaystyle V$ spanned by vectors $\displaystyle (1,2,0,1)$ and $\displaystyle (1,1,1,0)$, and $\displaystyle W_2$ is a subspace of $\displaystyle V$ spanned by $\displaystyle (2,3,1,1)$.

    Determine $\displaystyle dim(W_1+W_2)$ and $\displaystyle dim(W_1 \cap W_2)$.
    My other thread might be useful: http://www.mathhelpforum.com/math-he...subspaces.html

    Sifting $\displaystyle (1,2,0,1)$ and $\displaystyle (1,1,1,0)$ gives those two vectors. Hence $\displaystyle dim(W_1)=2$.

    Similarly $\displaystyle dim(W_2)=1$.

    Since $\displaystyle (1,2,0,1)+(1,1,1,0)=(2,3,1,1)$ we know that $\displaystyle W_2$ is a subspace of $\displaystyle W_1$.

    Therefore $\displaystyle dim(W_1 \cap W_2)=1$ and $\displaystyle dim(W_1+W_2)=2$.

    Is this it? This question was worth 10 marks but my answer is remarkably small!
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  2. #2
    MHF Contributor

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    Quote Originally Posted by Showcase_22 View Post
    My other thread might be useful: http://www.mathhelpforum.com/math-he...subspaces.html

    Sifting $\displaystyle (1,2,0,1)$ and $\displaystyle (1,1,1,0)$ gives those two vectors. Hence $\displaystyle dim(W_1)=2$.

    Similarly $\displaystyle dim(W_2)=1$.

    Since $\displaystyle (1,2,0,1)+(1,1,1,0)=(2,3,1,1)$ we know that $\displaystyle W_2$ is a subspace of $\displaystyle W_1$.

    Therefore $\displaystyle dim(W_1 \cap W_2)=1$ and $\displaystyle dim(W_1+W_2)=2$.

    Is this it? This question was worth 10 marks but my answer is remarkably small!
    your solution is correct.
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  3. #3
    Super Member Showcase_22's Avatar
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    your solution is correct.
    nifty!

    If you were doing this question, would you add in anything else?

    It seems a little small for 10 marks!
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