Definition

Let V be a vector space and
T : V → V a linear transformation.
A subspace W ⊆ V is invariant under T if:

T(W) ⊆ W

Meaning: whenever w ∈ W, applying T keeps you inside W.

Basic Examples

1) Trivial invariant subspaces

These exist for every linear operator:

• {0}
• The whole space V

2) Eigenspaces

If v is an eigenvector of T with eigenvalue
λ, then T v = λ v, and the span of v
is invariant. More generally, the eigenspace
E_λ = { v : T v = λ v } is invariant.

3) Upper-triangular matrix example

Let

T = [ [1, 1],
      [0, 1] ]

The subspace W = span{(1,0)} is invariant because
T(1,0) = (1,0).
But the y-axis is not invariant since
T(0,1) = (1,1) is not in span{(0,1)}.

Function Space Examples

4) Differential operator

Let T = d/dx.

• The space of all polynomials is invariant.
• The space of polynomials of degree ≤ n is invariant.
• The space of even functions is not invariant (derivative of even is odd).
• The space of odd functions is invariant under d²/dx².

5) Fourier / frequency subspaces

For periodic functions and T = d/dx, each Fourier mode
e^{ikx} spans an invariant subspace:

• The span of e^{ikx} for fixed k is invariant.
• This is why many PDEs “decouple” in Fourier space.

Quantum Mechanics Connection

In quantum mechanics, invariant subspaces often correspond to conserved sectors.
If a Hamiltonian respects a symmetry, the Hilbert space can decompose into invariant subspaces
that evolve independently.

Representation Theory View

• A representation is irreducible if it has no nontrivial invariant subspaces.
• Decomposing a space into invariant subspaces reveals “independent components” of an action.