Invariant subspace of linear mapping

Question

Let $T$ be a linear transformation on a complex vector space $V$ of dimension $4$ and let $\lambda_1,\lambda_2,\lambda_3$ be distinct eigenvalues of $T$.

Eigenvectors are: $\lambda_1\rightarrow \{v_1,v_2\}$, $\lambda_2\rightarrow \{v_3\}$, and $\lambda_3\rightarrow \{v_4\}$.

I want to find how many invariant subspaces I have.

I know that $V(\lambda_2)$ , $V(\lambda_3)$ , $V(\lambda_2)+V(\lambda_3)$ are distinct invariant subspaces.

Whats about the $V(\lambda_1)$ how many invariant subspaces I have?

If $\alpha\in\mathbb{C}$, then $\langle v_1 + \alpha v_2\rangle$ is invariant subspace?

Answer 1

Hints: The one-dimensional invariant subspaces: $L(v_3)$, $L(v_4)$, $L(v_2)$, $L(v_1+\alpha v_2)$, $\alpha\in\mathbb{C}$.

The two-dimensional invariant subspaces: $V(\lambda_1)$, $L(w,v_3)$, $L(w,v_4)$, $L(v_3,v_4)$, $w\in V(\lambda_1)$, $w\neq0$.

The three-dimensional invariant subspaces: $L(v_1,v_2,v_3)$, $L(v_1,v_2,v_4)$, $L(w,v_3,v_4)$, $w\in V(\lambda_1)$, $w\neq0$.

$L(w_1,\ldots,w_k)$ is a linear envelope of vectors $w_1,\ldots,w_k$, i.e. the subspace spanned by $w_1,\ldots,w_k$.