Editing Kernel (algebra) (section)

=== Linear maps ===
{{Main|Kernel (linear algebra)}}
Let ''V'' and ''W'' be [[vector space]]s over a [[Field (mathematics)|field]] (or more generally, [[module (mathematics)|modules]] over a [[Ring (mathematics)|ring]]) and let ''T'' be a [[linear map]] from ''V'' to ''W''. If '''0'''<sub>''W''</sub> is the [[zero vector]] of ''W'', then the kernel of ''T'' (or null space<ref name="Axler Kernel Examples"/>) is the [[preimage]] of the [[zero space|zero subspace]] {'''0'''<sub>''W''</sub>}; that is, the [[subset]] of ''V'' consisting of all those elements of ''V'' that are mapped by ''T'' to the element '''0'''<sub>''W''</sub>. The kernel is usually denoted as {{nowrap|ker ''T''}}, or some variation thereof:
: <math> \ker T = \{\mathbf{v} \in V : T(\mathbf{v}) = \mathbf{0}_{W}\} . </math>

Since a linear map preserves zero vectors, the zero vector '''0'''<sub>''V''</sub> of ''V'' must belong to the kernel. The transformation ''T'' is injective if and only if its kernel is reduced to the zero subspace.<ref>{{harvnb|Axler|p=60}}</ref>

The kernel ker ''T'' is always a [[linear subspace]] of ''V''.<ref name="Dummit Dimension">{{harvnb|Dummit|Foote|2004|p=413}}</ref> Thus, it makes sense to speak of the [[quotient space (linear algebra)|quotient space]] {{nowrap|''V'' / (ker ''T'')}}. The first isomorphism theorem for vector spaces states that this quotient space is [[natural isomorphism|naturally isomorphic]] to the [[image (function)|image]] of ''T'' (which is a subspace of ''W''). As a consequence, the [[dimension (linear algebra)|dimension]] of ''V'' equals the dimension of the kernel plus the dimension of the image.<ref name="Dummit Dimension"/>