Definition of eigenspace of a linear transformation

Definition: Let (T: V o V) be a linear transformation and (\lambda) an eigenvalue of (T). The eigenspace corresponding to (\lambda) is:
[E_\lambda = {\mathbf{v} \in V : T(\mathbf{v}) = \lambda \mathbf{v}} = \ker(T - \lambda I)]

The eigenspace is the set of all eigenvectors corresponding to (\lambda), together with the zero vector.

Example: For (T(x,y,z) = (2x, 2y, 3z)), the eigenspace for (\lambda = 2) is (E_2 = {(x, y, 0) : x, y \in \mathbb{R}}) (the xy-plane), and (E_3 = {(0, 0, z) : z \in \mathbb{R}}) (the z-axis).