We know that the improper subspaces of a vector space $V$ , $\{0\}$ and $V$ itself, are invariant under the linear operator $T$ and as we are always in search of proper invariant subspaces. (where $T \in L (V)$ and $dim (V)=n$.)
Now the eigenspace $W$ corresponding to some eigenvalue $c$ of $T$ is also invariant under $T$ $$W = \{x \in \mathbb{R}^n : Tx = cx\}$$ Now here is my question: When we talk about invariant subspaces we know that subspaces contain necessarily a zero vector but here $\{0\}$ doesn't belong to $W$ because eigenvectors are meaningful for non-zero vectors then how could be it a subspace.