Kernel of Quotient Mapping

Theorem
Let $V$ be a vector space.

Let $M$ be a subspace of $V$.

Let $Q: V \to V / M$ be the quotient mapping.

Then $\ker Q = M$, where $\ker Q$ is the kernel of $Q$.

Proof
For $v \in V$, we have that:


 * $v \in \ker Q$


 * $v + M = 0 + M$

That is, $v \in M$.

Hence $\ker Q = M$.