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Math Help - Vector Space question

  1. #1
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    Vector Space question

    Hi,

    Could someone explain to me why the set of all elements of R^3 with the first component of 1 is not a real vector space? I'm lost on this one. Thanks,

    Kim
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi
    Quote Originally Posted by Kim Nu View Post
    Could someone explain to me why the set of all elements of R^3 with the first component of 1 is not a real vector space? I'm lost on this one. Thanks,
    It can't be a vector space because this set is not closed under addition :

    <br />
\begin{pmatrix}<br />
1\\<br />
0\\<br />
0<br />
\end{pmatrix}<br />
+\begin{pmatrix}<br />
1\\<br />
0\\<br />
0<br />
\end{pmatrix}=\begin{pmatrix}<br />
2\\<br />
0\\<br />
0<br />
\end{pmatrix}<br />
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Kim Nu View Post
    Hi,

    Could someone explain to me why the set of all elements of R^3 with the first component of 1 is not a real vector space? I'm lost on this one. Thanks,

    Kim
    To be a vector space all linear combinations of elements would have to be in the set, but as the sum of any two elements is not in the set it is not a vector space.

    RonL
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  4. #4
    o_O
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    Consider vectors: \bold{u} = (1, u_{2}, u_{3}) and \bold{v} = (1, v_{2}, v_{3}).

    One of the axioms state that if u and v are vectors in your vector space, then so should u + v. However:
    \bold{u} + \bold{v} = (2, u_{2} + v_{2}, u_{3} + v_{3})

    which does not belong in your vector space.
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