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.