Definition of invariant subspace of a linear transformation.

Definition: A subspace (W \subseteq V) is invariant (or T-invariant) under a linear transformation (T: V o V) if:
[T(\mathbf{w}) \in W \quad ext{for all } \mathbf{w} \in W]

That is, applying (T) to any vector in (W) keeps the result within (W).

Examples of invariant subspaces:

• ({\mathbf{0}}) and (V) are always invariant (the trivial invariant subspaces)
• Every eigenspace (E_\lambda) is invariant: if (\mathbf{v} \in E_\lambda), then (T(\mathbf{v}) = \lambda\mathbf{v} \in E_\lambda)
• The kernel and range of (T) are invariant under (T)
• For a rotation in (\mathbb{R}^3) about the z-axis, the z-axis and the xy-plane are both invariant

Invariant subspaces allow a transformation to be "block-diagonalized" by decomposing (V) into smaller pieces.