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:

$v \in \ker Q$

if and only if

$v + M = 0 + M$

That is, if and only if $v \in M$.

Hence $\ker Q = M$.

$\blacksquare$

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): Appendix $\text{A}$ Preliminaries: $\S 1.$ Linear Algebra