Why invariant subspaces are important?

Question

My teacher says that to study a linear transformation $T:V\rightarrow V$ is equivalent to study $T|_{U_{i}}$, where $U_i$ are invariant subspaces of $V$.

Why so? I cant understand. Please help to understand

Answer 1

For instance:

• $T$ is invertible if and only if each $T|_{U_i}$ is invertible;
• $T$ is diagonalizable if and only if each $T|_{U_i}$ is diagonalizable.

But the $U_i$'s are smaller spaces. Therefore, it easier (in general) to study each $T|_{U_i}$ than the whole transformation $T$.