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
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.