**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.