Let $T$ be a linear operator on $V$.If every subspace of $V$ is invariant under $T$,then $T$ is a scalar multiple of the identity operator.
This problem is from Hoffman Kunze form chapter Invariant Subspace.
Let $\alpha_1$ be a non zero vector then $T(\alpha_1)$ is in the subspace generated by $\alpha_1$ since every subspace is invariant under $T$.
hence $T(\alpha_1)$ = $\lambda_1\alpha_1$
Now considering another element $\alpha_2$ from the complement of the subspace generated by $\alpha_1$.
Similarly, $T(\alpha_2)$ = $\lambda_2\alpha_2$
Now $T(\alpha_1+\alpha_2)$ $=$ $T(\alpha_1)$ $+$ $T(\alpha_2)$ $=$ $\lambda_1\alpha_1$ + $\lambda_2\alpha_2$
But I cannot proceed further.
What about this theorem when $V$ is inifinte dimensional?