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Thread: Linear subspace

  1. #1
    Junior Member
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    Aug 2009
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    Linear subspace

    Hi everybody,

    I don't know how to show that the following sets aren't linear subspaces:

    $\displaystyle F_1$={$\displaystyle (x;y;z)/x+y+z=1$}
    $\displaystyle F_2$={$\displaystyle (x;y;z)/xy=0$}
    $\displaystyle F_3$={$\displaystyle (x;y;z)/x^2=z^2$}
    ...

    Can you help me please????
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  2. #2
    MHF Contributor Swlabr's Avatar
    Joined
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    Quote Originally Posted by lehder View Post
    Hi everybody,

    I don't know how to show that the following sets aren't linear subspaces:

    $\displaystyle F_1$={$\displaystyle (x;y;z)/x+y+z=1$}
    $\displaystyle F_2$={$\displaystyle (x;y;z)/xy=0$}
    $\displaystyle F_3$={$\displaystyle (x;y;z)/x^2=z^2$}
    ...

    Can you help me please????
    To show that $\displaystyle U$ is a subspace of a vector space $\displaystyle V$ where $\displaystyle U \subset V$ you have two things to show,

    that it is closed under scalar multiplication (for $\displaystyle u \in U$ you need to show that $\displaystyle \alpha u \in U$),

    that it is closed under addition (for $\displaystyle u, v \in U$ you need to show that $\displaystyle u+v \in U$).

    You need to show these two things as a subspace is a vector space contained in a bigger space, and this bigger space lends the smaller space the other properties. So all you need to prove is these two things.

    So, for $\displaystyle F_1$ you need to show that $\displaystyle (x_1+x_2)+(y_1+y_2)+(z_1+z_2) = 1$ and that $\displaystyle \alpha x_1 + \alpha y_1 + \alpha z_1=1$ for $\displaystyle x_1+y_1+z_1=1$ and $\displaystyle x_2+y_2+z_2=1$.

    Do you understand why this is what you need to show?

    Now, does this hold?

    The other two questions can be attacked similarly.
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