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 $W=span(e_1,e_2)$?
Best Answer
It is easy to see that $T(e_1)=e_3 \in W$ and that $T(e_3)=e_1+e_3 \in W.$
This gives
$$T(W)=W,$$
hence $W$ is a $T-$ invariant subspace.