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Thread: 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 : $\displaystyle K$ is at the same time the subspace spanned by $\displaystyle M$ and $\displaystyle x$ and the subspace spanned by $\displaystyle M$ and $\displaystyle y$... but these subspaces may be different !

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

    We then have:

    $\displaystyle
    \begin{aligned}
    \mathbb{K}_x=&\;span\{\mathbb{M},x\}\\
    \mathbb{K}_y=&\;span\{\mathbb{M},y\}
    \end{aligned}
    $

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

    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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