Is the kernel of a direct sum of linear maps the direct sum of kernels?

Question

If we have two linear maps $\phi,\psi$ between finite dimensional vectors spaces, is it true that the kernel of $$ \phi \oplus \psi = \begin{bmatrix} \phi & 0 \\ 0 & \psi \end{bmatrix} $$ is equal to $\ker \phi \oplus \ker \psi$? If this is not true in general, is it at least true that if $\phi,\psi$ are injective (have trivial kernel) then $\phi \oplus \psi$ is injective?

Answer 1

$(\phi \oplus \psi)(x,y)=(0,0) \iff \phi(x)=0$ and $ \psi(y)=0 \iff (x,y) \in ker(\phi) \oplus \ker (\psi) $.