Vector Space question

Kim Nu · #1 $Old$ June 20th, 2008, 01:34 PM

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 · #2 $Old$ June 20th, 2008, 01:43 PM

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 :

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

CaptainBlack · #3 $Old$ June 20th, 2008, 01:43 PM

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

o_O · #4 $Old$ June 20th, 2008, 01:44 PM

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.