Houjun Liu

null space

The Null Space, also known as the kernel, is the subset of vectors which get mapped to \(0\) by some Linear Map.


Some linear map \(T \in \mathcal{L}(V,W)\)


The subset of \(V\) which \(T\) maps to \(0\) is called the “Null Space”:

\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.


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 \(\{0\}\)