---
title: "null space"
type: concept
related: [Zero, Linear Map, Subspace, Null Space, Closed]
source: https://www.jemoka.com/posts/kbhnull_space/
confidence: high
status: active
---

The Null Space, also known as the kernel, is the subset of vectors which get mapped to \(0\) by some Linear Map.
constituents Some linear map \(T \in \mathcal{L}(V,W)\)
requirements The subset of \(V\) which \(T\) maps to \(0\) is called the &ldquo;Null Space&rdquo;:
\begin{equation} null\ T = \{v \in V: Tv = 0\} \end{equation}
additional information the null space is a subspace of the domain It should probably not be a surprise, given a Null Space is called a Null Space, that the Null Space is a subspace of the domain.
zero As linear maps take \(0\) to \(0\), \(T 0=0\) so \(0\) is in the Null Space of \(T\).
closure under addition We have that:
\begin{equation} 0+0 = 0 \end{equation}
so by additivity of the Linear Maps the map is closed under addition.
closure under scalar multiplication By homogeneity of linear maps, the same of the above holds.
This completes the subspace proof, making \(null\ T\) a subspace of the domain of \(T\), \(V\). \(\blacksquare\)
the null space of the zero map is just the domain I mean duh. The zero map maps literally everything to zero.
Injectivity IFF implies that null space is \(\{0\}\) See injectivity IFF implies that null space is \(\{0\}\)
