Finding Invariant Subspaces

Find all nontrivial subspaces of the three-dimensional space that are invariant with respect to the linear operator given by the matrix

Find, in the three-dimensional linear space $\mathbb {R}^{3}$, all subspaces that are invariant with respect to each of the two operators given by the matrices

Find all nontrivial invariant subspaces of the operator $\mathbf{A}: \mathbb {R}^{3} \rightarrow \mathbb {R}^{3}$ that maps an arbitrary vector $x \in \mathbb {R}^{3}$ to $[x, a]([\cdot , \cdot ]$ is the cross product in $\mathbb {R}^{3}, a$ is a fixed vector).

Find all invariant subspaces of the differentiation operator $\frac{d}{d x}$ in the space of polynomials $\mathbb {k}_{n}[x]$, where the field $\mathbb {k}$ has characteristic zero.

Find all invariant subspaces of the differentiation operator $\frac{d}{d x}$ in the subspace of functions on the real line spanned by the functions $e^{\lambda_{1} x}, \ldots , e^{\lambda_{k} x}, \lambda_{i} \neq \lambda_{j}$ for $i \neq j$. What is the total number of such subspaces?

Find the $(n-1)$-dimensional invariant subspaces of the operator $\mathrm{A}: \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ given by the matrix: