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 “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.

#### 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.