# Linear Algebra: Vector Space!

• Jan 30th 2009, 08:37 PM
Vedicmaths
Linear Algebra: Vector Space!
Hello,

I need help in this linear Algebra problem...

Prove that W1 = {(a1, a2, a3....a3) E F^n: a1+a2+a3+...+an = 0} is a subspace of F^n, but
W2 = {(a1, a2, a3....a3) E F^n: a1+a2+a3+...+an = 0} is not...

Thanks for the help in advance!
• Jan 31st 2009, 10:48 AM
clic-clac
What is written leads to $\displaystyle W_1=W_2$, perhaps we should replace $\displaystyle =$ by $\displaystyle \neq$ in the definition of $\displaystyle W_2$. If so, since a subspace of a vector space $\displaystyle E$ must contain $\displaystyle 0_{E}$ and that $\displaystyle 0+...+0=0,$ we can conclude that $\displaystyle W_2$ doesn't contain $\displaystyle (0,...,0)$ and isn't a vector space.

By the way, $\displaystyle W_1$ contains $\displaystyle (0,...,0).$ So the only thing we have to prove now is that for every $\displaystyle \lambda ,\mu\in k$ (if $\displaystyle F$ is a $\displaystyle k$-vector space), $\displaystyle (a_1,...,a_n),(b_1,...,b_n)\in F^n\Rightarrow \lambda(a_1,...,a_n)+\mu(b_1,...,b_n)\in F^n.$

Let $\displaystyle (a_1,...a_n)$ and $\displaystyle (b_1,...,b_n)$ be two elements of $\displaystyle F^n,$ since $\displaystyle \lambda(a_1,...,a_n)+\mu(b_1,...,b_n)=(\lambda a_1+\mu a_1,...,\lambda a_n+\mu a_n),$

$\displaystyle \lambda(a_1,...,a_n)+\mu(b_1,...,b_n)\in F^n$
$\displaystyle \Leftrightarrow \lambda a_1+\mu b_1+...+\lambda a_n+\mu b_n)=0$
$\displaystyle \Leftrightarrow \lambda(\underbrace{a_1+...+a_n}_{=0})+\mu(\underb race{b_1+...+b_n}_{=0})=0$
$\displaystyle \Leftrightarrow \lambda .0+\mu .0=0$
$\displaystyle \Leftrightarrow 0=0$

Therefore $\displaystyle \lambda(a_1,...,a_n)+\mu(b_1,...,b_n)\in F^n$, and $\displaystyle W_1$ is a subspace of $\displaystyle F^n$