Editing Kernel (linear algebra) (section)

==Properties==
[[File:Kernel and image of linear map.svg|thumb|300px|Kernel and image of a linear map {{mvar|L}} from {{mvar|V}} to {{mvar|W}}]]
The kernel of {{mvar|L}} is a [[linear subspace]] of the domain {{mvar|V}}.<ref name="textbooks">Linear algebra, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in {{harvnb|Lay|2005}}, {{harvnb|Meyer|2001}}, and Strang's lectures.</ref><ref name=":0" />
In the linear map <math>L : V \to W,</math> two elements of {{mvar|V}} have the same [[Image (mathematics)|image]] in {{mvar|W}} if and only if their difference lies in the kernel of {{mvar|L}}, that is,
<math display=block>L\left(\mathbf{v}_1\right) = L\left(\mathbf{v}_2\right) \quad \text{ if and only if } \quad L\left(\mathbf{v}_1-\mathbf{v}_2\right) = \mathbf{0}.</math>

From this, it follows by the [[Isomorphism_theorems#First_isomorphism_theorem|first isomorphism theorem]] that the image of {{mvar|L}} is [[Vector space isomorphism|isomorphic]] to the [[Quotient space (linear algebra)|quotient]] of {{mvar|V}} by the kernel:
<math display=block>\operatorname{im}(L) \cong V / \ker(L).</math>
{{anchor|nullity}}In the case where {{mvar|V}} is [[finite-dimensional]], this implies the [[rank–nullity theorem]]:
<math display=block>\dim(\ker L) + \dim(\operatorname{im} L) = \dim(V).</math>
where the term {{em|{{visible anchor|rank}}}} refers to the dimension of the image of {{mvar|L}}, <math>\dim(\operatorname{im} L),</math> while ''{{em|{{visible anchor|nullity}}}}'' refers to the dimension of the kernel of {{mvar|L}}, <math>\dim(\ker L).</math><ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|url=http://mathworld.wolfram.com/Rank-NullityTheorem.html|title=Rank-Nullity Theorem|website=mathworld.wolfram.com|language=en|access-date=2019-12-09}}</ref> 
That is,
<math display=block>\operatorname{Rank}(L) = \dim(\operatorname{im} L) \qquad \text{ and } \qquad \operatorname{Nullity}(L) = \dim(\ker L),</math>
so that the rank–nullity theorem can be restated as 
<math display=block>\operatorname{Rank}(L) + \operatorname{Nullity}(L) = \dim \left(\operatorname{domain} L\right).</math>

When {{mvar|V}} is an [[inner product space]], the quotient <math>V / \ker(L)</math> can be identified with the [[orthogonal complement]] in {{mvar|V}} of <math>\ker(L)</math>. This is the generalization to linear operators of the [[row space]], or coimage, of a matrix.