The idea is to create a RESTRICTED space within the given vector space.

E.g. Within R3

\R^3: \{ \vec{i}, \vec{j}, \vec{k} \}

we can define a space that is far more limited:

\X^3: \{ \vec{x}, \vec{0}, \vec{-x} \}

This would be a subspace of R3 (it is an R2 space).

An Invariant Subspace

is one that retains the entire subspace under a transformation. In theory, when going from V to V (under T), there is no guarantee where a particular element of W (subset of V) will end up. It could end up anywhere within V. However, if it ends up exactly inside the same subset W, then we say that W is an invariant subspace of V.

An example of an Invariant Transformation Subspace

Consider R3 and consider the transformation of rotating around the z-axis (by an angle theta).

Any point on the X-Y plane will END UP again on the X-Y plane under such a rotation.

We say that the X-Y plane (z=0) is  invariant under the angle rotation transformation.

The connection with Quantum Mechanics?