The eigenspace of a linear transformation is a nontrivial subspace.

Theorem: For any eigenvalue (\lambda) of a linear transformation (T: V o V), the eigenspace (E_\lambda = \ker(T - \lambda I)) is a nontrivial subspace of (V) (i.e., (E_\lambda
eq {\mathbf{0}})).

Proof: Since (\lambda) is an eigenvalue, by definition there exists a nonzero vector (\mathbf{v}) such that (T(\mathbf{v}) = \lambda \mathbf{v}), which means ((T - \lambda I)(\mathbf{v}) = \mathbf{0}), so (\mathbf{v} \in \ker(T - \lambda I) = E_\lambda). Thus (E_\lambda) contains at least one nonzero vector.

Since the kernel of any linear transformation is a subspace, (E_\lambda) is a subspace. Combined with the existence of a nonzero vector, it is a nontrivial subspace.

Consequence: The dimension of (E_\lambda) (called the geometric multiplicity of (\lambda)) is at least 1.