Invariant subspace of $R^3$ Let $T:R^3→R^3$ be the linear operator defined by 
$$T(\begin{bmatrix}a\\b\\c\end{bmatrix})=\begin{bmatrix}b+c\\2b\\a-b+c\end{bmatrix}$$
Show that $W=span(e_1,e_3)$ is a T-invariant subspace of $R^3$.
Let $\alpha={e_1,e_3} $ be ordered basis for W and $\beta={e_1,e_2,e_3}$ be ordered basis for $R^3=V$.
(In my textbook's example, W was $W=span({e_1,e_2})$ and the $T_W:W→W,\begin{bmatrix}s\\t\\0\end{bmatrix}→\begin{bmatrix}t\\-s\\0\end{bmatrix}$) So my question is how can I show that W is a T-invariant subspace? And also how can I write matrices like $W=span(e_1,e_2)$?

You see that, being $a,c\in \mathbb{R}$,
$$T(a\cdot e_1 + c\cdot e_3) = \begin{pmatrix}c\\0\\a+c\end{pmatrix}$$
Clearly, $$\begin{pmatrix}c\\0\\a+c\end{pmatrix}\in \text{span}(e_1,e_3)=W$$ because it equals $(c\cdot e_1 + (a+c)\cdot e_3)$, so it's a $T$-invariant subspace by the definition of invariant subspace $(T(W)\subseteq W)$.
