Find all T-invariant subspaces of a linear transformation T Let $T$ be the linear transformation on $\mathbb{R}^2$ which is represented by the matrix $\left(\begin{array}{cc}1 & \lambda\\0 &1\end{array}\right)$, $\lambda\ne0$ in the standard basis $\{(1,0),(0,1)\}$ of $\mathbb{R}^2$. Find all $T$-invariant subspaces.
I have no idea how to start solving exercises as the above. 
 A: It's clear that $\operatorname{span}((1,0))$ is a $T$-invariant subspace since $$T(1,0)=(1,0)$$ and obviously $\Bbb R^2$ and $\{(0,0)\}$ are two $T$-invariant.
Now if there's another $T$-subspace with dimension $1$ so let $v$ a vector that span this subspace and then
$$\Bbb R^2=\operatorname{span}(1,0)\oplus \operatorname{span}(v)$$
hence we see that the matrix of $T$ in the basis $((1,0),v)$ is diagonal which's impossible.
A: Suppose $W$ is a T-invariant subspace of $\mathbb{R}^2$. Then $dim(W) \in \{0,1,2\}$. That is, or the T-invariant subspace is $W=\{0\}$ or $W=\mathbb{R}^2$ or it has dimension 1. So, in the case that $dim(W)=1$, I'll prove that, necessarily, $W \subset V_{\lambda}$ (W is in a auto-space).
Suppose then $w \in W$ with $W$ T-invariante and $dim(W)=1$, then $T(w) \in W$ and hence $T(w)=\lambda\cdot w$ (for some $\lambda$ in the field, because $dim(W)=1$) $\Rightarrow w\in V_{\lambda} \Rightarrow W \subset V_{\lambda}$.
Particulary, how the only auto-space of T has dimension 1, it follows that $W = V_{\lambda}$ and hence, the only T-invariant subspaces are those three.
