Answer to Question #339159 in Linear Algebra for Matt

Question

Show that if W consist of these vectors (a, b, c)€R³ for which a=2b then W is subspace of R³

Answer 1

$W$ consists of vectors $(2b,b,c)$, where $b,c\in{\mathbb{R}}$. We will use a subspace criterion to check that $W$ is a subspace of ${\mathbb{R}}^3$. Namely, we have to check the following:

1. $0\in W$.
2. $u+v\in W$ for all vectors $u,v\in W$.
3. $\alpha u\in W$ for any $\alpha\in{\mathbb{R}}$ and for an arbitrary vector $u\in W$.

It is obvious that $0\in W$. Take two arbitrary vectors $u=(2b,b,c)$ and $v=(2d,d,f)$ from $W$ with $b,c,d,f\in{\mathbb{R}}$. Then, $u+v=(2(b+d),b+d,c+f)$. As we can see, $u+v\in W$. Consider arbitrary vector $u=(2b,b,c)$ from $W$ with $b,c\in{\mathbb{R}}$ . Then, $\alpha u=(2\alpha b,\alpha b,c)\in W$ for all $\alpha\in{\mathbb{R}}$. Thus, we checked that $W$ is a subspace of $\mathbb{R}^3$.