Q:

How do i show V is a linear subspace if it's defined like this?

Let $V =\{(x,y,z) \in \mathbb R^3 : x+3y=3z\}$,and let $T :V \to \mathbb R^3$. be given by $T(x,y,z)=(x,y,z)\times(1,3,−3)$, the usual cross-product in $\mathbb R^3$.

How do i show that V is a linear subspace? I'm confused because I know this as a matrix. 

A:

To show that $V$ is a subspace, you have to show that for any vectors $u,v \in V$ and scalar $\alpha, \beta$, $\alpha u + \beta v \in V$ as well. (This is the definition, but in fact, showing that $\alpha u + v \in V$ turns out to be sufficient.)
So let $(a,b,c), (x,y,z) \in V$ and $\alpha \in \mathbb{R}$, and test whether $\alpha (a,b,c) + (x,y,z) \in V$, that is, whether it fulfils the definition given for $V$.

