Answer to Question #235670 in Linear Algebra for Amuj

Question

Let the vector space V=R^3 and W={(a,b,c);a+b+c=0} i.e. W consists of those vectors each with the property that the sum of its components is zero. Is W a subspace of V

Answer 1

Let us use the subspace criterion: a nonempty subset $W\subset V$ is a vector subspace of the space $V$ over the field $R$ , if and only if the following statements are true: $1) \forall u, v\in W (u+v)\in W;$ $2) \forall\alpha\in R \forall u \in W (\alpha u)\in W$ .

Let's check these statements for the case $V=R^3$  and $W=\{(a,b,c);a+b+c=0\}$

1) $\forall u(a;b;c), v(m;n;k)\in W$ $(u+v)=(a+m;b+n;c+k)$ and $a+m+b+n+c+k=a+b+c+m+n+k=0+0=0$ . So $(u+v)\in W$

2) $\forall\alpha\in R \forall u (a;b;c) \in W$ $(\alpha u)= (\alpha a; \alpha b; \alpha c)$ and $\alpha a+\alpha b+\alpha c =\alpha (a+b+c)=0$ .

So $(\alpha u)\in W$ . So $W$ is a vector subspace of the space $V$