Verify that the following sequence of vector spaces is an exact sequence

Question

The sequence is $0\rightarrow V_1 \rightarrow V_1\oplus V_2\rightarrow V_2\rightarrow 0$ and $l:V_1\rightarrow V_1\oplus V_2$ is an inclusion and $p:V_1\oplus V_2 \rightarrow 0$ is a projection.

It is not clear to me why $\ker(V_2\rightarrow 0)=\mathrm{im}(p)$ and also why $\ker(P)$ might be equal to $\mathrm{im}(l)$.

Answer 1

Explicitly, we want to see that:

• $l: V_1 \to V_1 \oplus V_2$ is injective;
• $\ker(p: V_1 \oplus V_2 \to V_2) = {\rm im}(l: V_1 \to V_1\oplus V_2)$;
• $p: V_1\oplus V_2 \to V_2$ is surjective.

These assertions say that the sequence is exact in $V_1$, $V_1 \oplus V_2$ and $V_2$ respectively. The first and last claims are immediate. Let's check the second.

Let $(v_1,v_2) \in \ker (p)$, so that $p(v_1,v_2) = v_2 = 0$. Then $(v_1,v_2) = (v_1, 0) = l(v_1) \in {\rm im}(l)$. On the other hand, if $l(v_1) \in {\rm im}(l)$, we have $p(l(v_1)) = p(v_1,0) = 0$ and $l(v_1) \in \ker(p)$, which ends the verification.