Editing Linear map (section)

==Kernel, image and the rank–nullity theorem==
{{Main|Kernel (linear algebra)|Image (mathematics)|Rank of a matrix}}
If <math display="inline">f: V \to W</math> is linear, we define the [[kernel (linear operator)|kernel]] and the [[image (mathematics)|image]] or [[Range of a function|range]] of <math display="inline">f</math> by
<math display="block">\begin{align}
    \ker(f) &= \{\,\mathbf x \in V: f(\mathbf x) = \mathbf 0\,\} \\
    \operatorname{im}(f) &= \{\,\mathbf w \in W: \mathbf w = f(\mathbf x), \mathbf x \in V\,\}
\end{align}</math>

<math display="inline">\ker(f)</math> is a [[Linear subspace|subspace]] of <math display="inline">V</math> and <math display="inline">\operatorname{im}(f)</math> is a subspace of <math display="inline">W</math>.  The following [[dimension]] formula is known as the [[rank–nullity theorem]]:<ref>{{harvnb|Horn|Johnson|2013|loc=0.2.3 Vector spaces associated with a matrix or linear transformation, p. 6}}</ref>
<math display="block">\dim(\ker( f )) + \dim(\operatorname{im}( f )) = \dim( V ).</math>

The number <math display="inline">\dim(\operatorname{im}(f))</math> is also called the [[rank of a matrix|rank]] of <math display="inline">f</math> and written as <math display="inline">\operatorname{rank}(f)</math>, or sometimes, <math display="inline">\rho(f)</math>;<ref name=":0">{{Harvard citation text|Katznelson|Katznelson|2008}} p. 52, § 2.5.1</ref><ref name=":1">{{Harvard citation text|Halmos|1974}} p. 90, § 50</ref> the number <math display="inline">\dim(\ker(f))</math> is called the [[Kernel (matrix)#Subspace properties|nullity]] of <math display="inline">f</math> and written as <math display="inline">\operatorname{null}(f)</math> or <math display="inline">\nu(f)</math>.<ref name=":0" /><ref name=":1" /> If <math display="inline">V</math> and <math display="inline">W</math> are finite-dimensional, bases have been chosen and <math display="inline">f</math> is represented by the matrix <math display="inline">A</math>, then the rank and nullity of <math display="inline">f</math> are equal to the rank and nullity of the matrix <math display="inline">A</math>, respectively.