Editing Rank–nullity theorem (section)

==Stating the theorem==

===Linear transformations===

Let <math>T : V \to W</math> be a linear transformation between two vector spaces where <math>T</math>'s domain <math>V</math> is finite dimensional. Then
<math display=block>\operatorname{rank}(T) ~+~ \operatorname{nullity}(T) ~=~ \dim V,</math>
where <math display=inline>\operatorname{rank}(T)</math> is the [[rank (linear algebra)|rank]] of <math>T</math> (the [[Dimension (vector space)|dimension]] of its [[Image (mathematics)|image]]) and <math>\operatorname{nullity}(T)</math> is the [[Nullity (linear algebra)|nullity]] of <math>T</math> (the dimension of its [[Kernel (linear algebra)|kernel]]).  In other words,
<math display=block>\dim (\operatorname{Im} T) + \dim (\operatorname{Ker} T) = \dim (\operatorname{Domain}(T)).</math>
This theorem can be refined via the [[splitting lemma]] to be a statement about an [[isomorphism]] of spaces, not just dimensions. Explicitly, since <math> T </math> induces an isomorphism from <math>V / \operatorname{Ker} (T)</math> to <math>\operatorname{Im} (T),</math> the existence of a basis for <math> V </math> that extends any given basis of <math>\operatorname{Ker}(T)</math> implies, via the splitting lemma, that <math>\operatorname{Im}(T) \oplus \operatorname{Ker}(T) \cong V.</math> Taking dimensions, the rank–nullity theorem follows.

=== Matrices ===
Linear maps can be represented with [[Matrix (mathematics)|matrices]]. More precisely, an <math>m\times n</math> matrix {{mvar|M}} represents a linear map <math>f:F^n\to F^m,</math> where <math>F</math> is the underlying [[field (mathematics)|field]].<ref>{{Harvard citation text|Friedberg|Insel|Spence|2014}} pp. 103-104, §2.4, Theorem 2.20</ref> So, the dimension of the domain of <math>f</math> is {{mvar|n}}, the number of columns of {{mvar|M}}, and the rank–nullity theorem for an <math>m\times n</math> matrix {{mvar|M}} is
<math display=block>\operatorname{rank}(M) + \operatorname{nullity}(M) = n.</math>