Row and column spaces: Difference between revisions

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In [[linear algebra]], the '''column space''' (also called the '''range''' or [[Image (mathematics)|'''image''']]) of a [[matrix (mathematics)|matrix]] ''A'' is the [[Linear span|span]] (set of all possible [[linear combination]]s) of its [[column vector]]s. The column space of a matrix is the [[image (mathematics)|image]] or [[range (mathematics)|range]] of the corresponding [[matrix transformation]].

Let <math>\mathbb{F}</math> be a [[field (mathematics)|field]]. The column space of ana ''m''&#8239;&times;&#8239;''n'' matrix with components from <math>\mathbb{F}</math> is a [[linear subspace]] of the [[Examples of vector spaces#Coordinate space|''m''-space]] <math>\mathbb{F}^m</math>. The [[dimension (linear algebra)|dimension]] of the column space is called the [[rank (linear algebra)|rank]] of the matrix and is at most min(''m'',&#8239;''n'').<ref name="ReferenceA">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 Lay 2005, Meyer 2001, and Strang 2005.</ref> A definition for matrices over a [[ring (mathematics)|ring]] <math>\mathbb{K}</math> [[#For matrices over a ring|is also possible]].

The '''row space''' is defined similarly.