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Thread: Set proof

  1. #1
    Senior Member Danneedshelp's Avatar
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    Set proof

    Q: Prove that if $\displaystyle S_{1}$ is a nonempty subset of the finite set $\displaystyle S_{2}$, and $\displaystyle S_{1}$ is linearly dependent, then so is $\displaystyle S_{2}$

    A: By definition $\displaystyle S_{1}\subset{S_{2}}$ $\displaystyle \iff$ $\displaystyle \forall{\vec{v}}\in{U}$$\displaystyle (\vec{v}\in{S_{1}}$ $\displaystyle \Rightarrow$ $\displaystyle \vec{v}\in{S_{2}})$. Since we know every vector $\displaystyle \vec{v_{i}}\in{S_{1}}$ can be written as a linear combination of at least one vector $\displaystyle \vec{u}\in{S_{1}}$ and $\displaystyle S_{1}\subset{S_{2}}$ we are guaranteed $\displaystyle \vec{u}\in{S_2}$. So, we have shown exists a linear dependent vector in $\displaystyle S_{2}$.

    $\displaystyle \therefore$ $\displaystyle S_{2}$ must also be linearly dependent.

    Have I shown enough work to prove my claim?
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    Linear dependence

    Hello Danneedshelp
    Quote Originally Posted by Danneedshelp View Post
    Q: Prove that if $\displaystyle S_{1}$ is a nonempty subset of the finite set $\displaystyle S_{2}$, and $\displaystyle S_{1}$ is linearly dependent, then so is $\displaystyle S_{2}$

    A: By definition $\displaystyle S_{1}\subset{S_{2}}$ $\displaystyle \iff$ $\displaystyle \forall{\vec{v}}\in{U}$$\displaystyle (\vec{v}\in{S_{1}}$ $\displaystyle \Rightarrow$ $\displaystyle \vec{v}\in{S_{2}})$. Since we know every vector $\displaystyle \vec{v_{i}}\in{S_{1}}$ can be written as a linear combination of at least one vector $\displaystyle \vec{u}\in{S_{1}}$ and $\displaystyle S_{1}\subset{S_{2}}$ we are guaranteed $\displaystyle \vec{u}\in{S_2}$. So, we have shown exists a linear dependent vector in $\displaystyle S_{2}$.

    $\displaystyle \therefore$ $\displaystyle S_{2}$ must also be linearly dependent.

    Have I shown enough work to prove my claim?
    This is OK, except that it's not every vector $\displaystyle \vec{v_{i}}\in{S_{1}}$, but $\displaystyle \exists\,\vec{v_{i}}\in{S_{1}}$.

    Grandad
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