Linear Algebra: Vector Space!

**Vedicmaths** · January 30th 2009, 08:37 PM

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!

**clic-clac** · January 31st 2009, 10:48 AM

What is written leads to $W_1=W_2$ , perhaps we should replace $=$ by $\neq$ in the definition of $W_2$ . If so, since a subspace of a vector space $E$ must contain $0_{E}$ and that $0+...+0=0,$ we can conclude that $W_2$ doesn't contain $(0,...,0)$ and isn't a vector space.

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

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

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

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

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