Editing Kernel (linear algebra) (section)

{{Short description|Vectors mapped to 0 by a linear map}}
{{other uses|Kernel (disambiguation)}}
[[File:Projection-on-diagonal.gif|thumb|300x300px|An example for a kernel- the linear operator <math> L : (x,y) \longrightarrow (x, x)</math> transforms all points on the <math> (x=0, y)</math> line to the zero point <math> (0,0)</math>, thus they form the kernel for the linear operator]]
In [[mathematics]], the '''kernel''' of a [[linear map]], also known as the '''null space''' or '''nullspace''', is the part of the [[Domain of a function|domain]] which is mapped to the [[Zero element#Additive identities|zero vector]] of the [[Codomain|co-domain]]; the kernel is always a [[linear subspace]] of the domain.<ref>{{Cite web|url=http://mathworld.wolfram.com/Kernel.html|title=Kernel|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-09}}</ref> That is, given a linear map {{math|''L'' : ''V'' → ''W''}} between two [[vector space]]s {{mvar|V}} and {{mvar|W}}, the kernel of {{mvar|L}} is the vector space of all elements {{math|'''v'''}} of {{mvar|V}} such that {{math|1=''L''('''v''') = '''0'''}}, where {{math|'''0'''}} denotes the [[zero vector]] in {{mvar|W}},<ref name=":0">{{Cite web|url=https://brilliant.org/wiki/kernel/|title=Kernel (Nullspace) {{!}} Brilliant Math & Science Wiki|website=brilliant.org|language=en-us|access-date=2019-12-09}}</ref> or more symbolically:
<math display="block">\ker(L) = \left\{ \mathbf{v} \in V \mid L(\mathbf{v})=\mathbf{0} \right\} = L^{-1}(\mathbf{0}).</math>