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Math Help - Subspace spanned by subspace

  1. #1
    Member Mollier's Avatar
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    Subspace spanned by subspace

    Hi,

    problem:
    Suppose that x and y are vectors and M is a subspace in a vector space V;
    let K be the subspace spanned by M and x, and let K be the subspace spanned by M and y.
    Prove that if y is in K but not in M, then x is in K.

    attempt:
    So, K is the set of all linear combinations of elements in M and the vector y.
    K is also the set of all linear combination of elements in M and the vector x.
    This definition of K makes me think that x is always in K.
    What am I missing here?

    Thanks!
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi,
    Quote Originally Posted by Mollier View Post
    problem:
    Suppose that x and y are vectors and M is a subspace in a vector space V;
    let K be the subspace spanned by M and x, and let K be the subspace spanned by M and y.
    Prove that if y is in K but not in M, then x is in K.
    It does not make sense : K is at the same time the subspace spanned by M and x and the subspace spanned by M and y... but these subspaces may be different !

    Here is what I think they meant :
    problem:
    Suppose that x and y are vectors and M is a subspace in a vector space V;
    let K_x be the subspace spanned by M and x, and let K_y be the subspace spanned by M and y.
    Prove that if y is in K_x but not in M, then x is in K_y.
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  3. #3
    Member Mollier's Avatar
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    Hi!

    We then have:

     <br />
\begin{aligned}<br />
\mathbb{K}_x=&\;span\{\mathbb{M},x\}\\<br />
\mathbb{K}_y=&\;span\{\mathbb{M},y\}<br />
\end{aligned}<br />

    If y\in\mathbb{K}_x but y\notin\mathbb{M} then x\in\mathbb{M} and therfore x\in\mathbb{K}_y.

    Is this any better?

    Thanks.
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  4. #4
    Member Mollier's Avatar
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    Gentle bump.
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