Block matrixes and invariant subspaces

Question

Let $À : L \to L, L = L_1 + L_2$. In my book it says if space $L$ is equal to the direct sum of $2$ invariant subspaces of $L$ then matrix $A$ (representation of linear operator $À$) can be divided into blocks like this \begin{bmatrix}A_1&0\\0&A_2\end{bmatrix} Where $A_1$ is restriction of $A$ on subspace $L_1$ and $A_2$ is that on $L_2$. So why couldn't we do the same if $L_1,L_2$ were just non invariant subspaces? In the proof, it says if $L_1$ is invariant then restriction of $À$ on $L_1$ will also be linear couldn't we say the same if $L_1$ were just a subspace?