Suppose $A, B$ are subspaces of $V$ . Define $T\colon A \times B → V $ by $F((x, y)) = x + y$.
I'm trying to show that $\ker(F) \cong A \cap B.$
I need to give an isomorphism. So I think I need show that it is linear, injective, and surjective. To show it's linear, can I just say that it is linear since $F$ is linear, or do I need to somehow prove it further? Also for injective and surjective, I think I need to find an inverse of this? I'm pretty confused, can someone maybe show me how to start off? I been at this for 2hours, please show me how the proof works.
I would start by proving $\mathrm{ker}(F) = \{(x,-x) \; | \; x \in A \cap B\}$. Once that is done, a good map to consider would be
$\varphi : A \cap B \to \mathrm{ker}(F) \; ; \; x \mapsto (x,-x).$
It has inverse $\psi : \mathrm{ker}(F) \to A \cap B \; ; \; (x,y) \mapsto x$, which you'll have to check. The fact that it has an inverse makes $\varphi$ a bijection. The fact that its inverse is a canonical projection (linear) makes $\varphi$ linear as well (recall that linear bijections have linear inverses and vice versa).