Vector Space question

**Kim Nu** · June 20th 2008, 11:34 AM

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

**flyingsquirrel** · June 20th 2008, 11:43 AM

Hi

Originally Posted by Kim Nu

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 :

$ \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix} +\begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix}=\begin{pmatrix} 2\\ 0\\ 0 \end{pmatrix} $

**CaptainBlack** · June 20th 2008, 11:43 AM

Originally Posted by Kim Nu

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

**o_O** · June 20th 2008, 11:44 AM

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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