Why are invariant subspaces called invariant?
Normally invariant in mathematics means something that doesn't change.
Suppose we have a linear map $T:V \to V$
Also suppose $P$ is a proper subspace of $Q$, and $Q$ is proper subspace of $V$ i.e. $P \subset Q \subset V$.
I mean $P \neq Q$ and $Q \neq V$.
Now if $T$ maps $Q$ to $P$, then $Q$ is called invariant subspace w.r.t. the linear map $T$.
But $Q$ is not really invariant in the usual sense, is it?
Because under the action of $T$ the subspace $Q$ changes (it "shrinks" so to say) to $P$, and $P \neq Q$.
Example:
$V = \mathbb{R}^3$
$T((x,y,z)) = (x,0,0)$
$P = \{(x,0,0)\}$
$Q = \{(x,y,0)\}$