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{{Merge |Row and column spaces|date=September 2013}}

[[File:Matrix Rows.svg|thumb|right|The row vectors of a [[matrix (mathematics)|matrix]]]]
In [[linear algebra]], the '''row space''' of a [[matrix (mathematics)|matrix]] is the set of all possible [[linear combination]]s of its [[row vector]]s. Let ''K'' be a [[field (mathematics)|field]] (such as [[real number|real]] or [[complex number|complex]] numbers). The row space of an ''m'' × ''n'' matrix with components from ''K'' is a [[linear subspace]] of the [[Examples of vector spaces #Coordinate space|''n''-space]] ''K''''n''. The [[dimension (linear algebra)|dimension]] of the row space is called the '''[[rank (linear algebra)|row rank]]''' of the matrix.<ref>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]] ''K'' (such as [[integer]]s) is also possible.<ref>A definition and certain properties for rings are the same with replacement of the "[[vector space|vector ''n''-space]]" ''K''''n'' with "left [[free module]]" and "linear subspace" with "[[submodule]]". For non-commutative rings this row space is sometimes disambiguated as ''left'' row space.</ref>

==Definition==
Let ''K'' be a [[field (mathematics)|field]] of [[scalar (mathematics)|scalars]]. Let ''A'' be an ''m'' × ''n'' matrix, with row vectors '''r'''1, '''r'''2, ... , '''r'''''m''. A [[linear combination]] of these vectors is any vector of the form
:<math>c_1 \mathbf{r}_1 + c_2 \mathbf{r}_2 + \cdots + c_m \mathbf{r}_m,</math>
where ''c''1, ''c''2, ... , ''cm'' are scalars. The set of all possible linear combinations of '''r'''1, ... , '''r'''''m'' is called the '''row space''' of ''A''. That is, the row space of ''A'' is the [[linear span|span]] of the vectors '''r'''1, ... , '''r'''''m''.

For example, if
:<math>A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 0 \end{bmatrix},</math>
then the row vectors are '''r'''1 = (1, 0, 2) and '''r'''2 = (0, 1, 0). A linear combination of '''r'''1 and '''r'''2 is any vector of the form
:<math>c_1 (1,0,2) + c_2 (0,1,0) = (c_1,c_2,2c_1).\,</math>
The set of all such vectors is the row space of ''A''. In this case, the row space is precisely the set of vectors (''x'', ''y'', ''z'') ∈ ''K''3 satisfying the equation ''z'' = 2''x'' (using [[Cartesian coordinates]], this set is a [[plane (mathematics)|plane]] through the origin in [[three-dimensional space]]).

For a matrix that represents a homogeneous [[system of linear equations]], the row space consists of all linear equations that follow from those in the system.

The column space of ''A'' is equal to the row space of ''A''T.

==Basis==
The row space is not affected by [[elementary row operations]]. This makes it possible to use [[row reduction]] to find a [[basis (linear algebra)|basis]] for the row space.

For example, consider the matrix
:<math>A = \begin{bmatrix} 1 & 3 & 2 \\ 2 & 7 & 4 \\ 1 & 5 & 2\end{bmatrix}.</math>
The rows of this matrix span the row space, but they may not be [[linearly independent]], in which case the rows will not be a basis. To find a basis, we reduce ''A'' to [[row echelon form]]:

'''r1''', '''r2''', '''r3''' represents the rows.
:<math>
\begin{bmatrix} 1 & 3 & 2 \\ 2 & 7 & 4 \\ 1 & 5 & 2\end{bmatrix} \underbrace{\sim}_{r_2-2r_1}
\begin{bmatrix} 1 & 3 & 2 \\ 0 & 1 & 0 \\ 1 & 5 & 2\end{bmatrix} \underbrace{\sim}_{r_3-r_1}
\begin{bmatrix} 1 & 3 & 2 \\ 0 & 1 & 0 \\ 0 & 2 & 0\end{bmatrix} \underbrace{\sim}_{r_3-2r_2}
\begin{bmatrix} 1 & 3 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{bmatrix} \underbrace{\sim}_{r_1-3r_2}
\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{bmatrix}.
</math>
Once the matrix is in echelon form, the nonzero rows are a basis for the row space. In this case, the basis is { (1, 3, 2), (2, 7, 4) }. Another possible basis { (1, 0, 2), (0, 1, 0) } comes from a further reduction.<ref name="example">The example is valid over real, [[rational number]]s, and other [[number field]]s. It is not necessarily correct over fields and rings with non-zero [[characteristic (algebra)|characteristic]].</ref>

This algorithm can be used in general to find a basis for the span of a set of vectors. If the matrix is further simplified to [[reduced row echelon form]], then the resulting basis is uniquely determined by the row space.

==Dimension==
{{main|Rank (linear algebra)}}
The [[dimension (linear algebra)|dimension]] of the row space is called the '''[[rank (linear algebra)|rank]]''' of the matrix. This is the same as the maximum number of linearly independent rows that can be chosen from the matrix. For example, the 3 × 3 matrix in the example above has rank two.<ref name="example"/>

The rank of a matrix is also equal to the dimension of the [[column space]]. The dimension of the [[null space]] is called the '''nullity''' of the matrix, and is related to the rank by the following equation:
:<math>\operatorname{rank}(A) + \operatorname{nullity}(A) = n,</math>
where ''n'' is the number of columns of the matrix ''A''. The equation above is known as the [[rank-nullity theorem]].

==Relation to the null space==
The [[null space]] of matrix ''A'' is the set of all vectors '''x''' for which ''A'''''x''' = '''0'''. The product of the matrix ''A'' and the vector '''x''' can be written in terms of the [[dot product]] of vectors:
:<math>A\mathbf{x} = \begin{bmatrix} \mathbf{r}_1 \cdot \mathbf{x} \\ \mathbf{r}_2 \cdot \mathbf{x} \\ \vdots \\ \mathbf{r}_m \cdot \mathbf{x} \end{bmatrix},</math>
where '''r'''1, ... , '''r'''''m'' are the row vectors of ''A''. Thus ''A'''''x''' = '''0''' if and only if '''x''' is [[orthogonal]] (perpendicular) to each of the row vectors of ''A''.

It follows that the null space of ''A'' is the [[orthogonal complement]] to the row space. For example, if the row space is a plane through the origin in three dimensions, then the null space will be the perpendicular line through the origin. This provides a proof of the [[rank-nullity theorem]] (see [[#Dimension|dimension]] above).

The row space and null space are two of the [[four fundamental subspaces]] associated with a matrix ''A'' (the other two being the [[column space]] and [[left null space]]).

==Relation to coimage==
If ''V'' and ''W'' are [[vector spaces]], then the [[kernel (linear algebra)|kernel]] of a [[linear transformation]] ''T'': ''V'' → ''W'' is the set of vectors '''v''' ∈ ''V'' for which ''T''('''v''') = '''0'''. The kernel of a linear transformation is analogous to the null space of a matrix.

If ''V'' is an [[inner product space]], then the orthogonal complement to the kernel can be thought of as a generalization of the row space. This is sometimes called the [[coimage]] of ''T''. The transformation ''T'' is one-to-one on its coimage, and the coimage maps [[isomorphism|isomorphically]] onto the [[image (mathematics)|image]] of ''T''.

When ''V'' is not an inner product space, the coimage of ''T'' can be defined as the [[quotient space (linear algebra)|quotient space]] ''V'' / ker(''T'').

==Notes==
{{reflist}}

==References==
{{see also|Linear algebra#Further reading}}

===Textbooks===
* {{Citation
| last = Axler
| first = Sheldon Jay
| date = 1997
| title = Linear Algebra Done Right
| publisher = Springer-Verlag
| edition = 2nd
| isbn = 0-387-98259-0
}}
* {{Citation
| last = Lay
| first = David C.
| date = August 22, 2005
| title = Linear Algebra and Its Applications
| publisher = Addison Wesley
| edition = 3rd
| isbn = 978-0-321-28713-7
}}
* {{Citation
| last = Meyer
| first = Carl D.
| date = February 15, 2001
| title = Matrix Analysis and Applied Linear Algebra
| publisher = Society for Industrial and Applied Mathematics (SIAM)
| isbn = 978-0-89871-454-8
| url = http://www.matrixanalysis.com/DownloadChapters.html
}}
* {{Citation
| last = Poole
| first = David
| date = 2006
| title = Linear Algebra: A Modern Introduction
| publisher = Brooks/Cole
| edition = 2nd
| isbn = 0-534-99845-3
}}
* {{Citation
| last = Anton
| first = Howard
| date = 2005
| title = Elementary Linear Algebra (Applications Version)
| publisher = Wiley International
| edition = 9th
}}
* {{Citation
| last = Leon
| first = Steven J.
| date = 2006
| title = Linear Algebra With Applications
| publisher = Pearson Prentice Hall
| edition = 7th
}}

==External links==
{{wikibooks|Linear Algebra/Column and Row Spaces}}
* {{MathWorld |title=Row Space |urlname=RowSpace}}
*{{aut|[[Gilbert Strang]]}}, [http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/detail/lecture10.htm MIT Linear Algebra Lecture on the Four Fundamental Subspaces] at Google Video, from [[MIT OpenCourseWare]]

{{linear algebra}}

[[Category:Linear algebra]]
[[Category:Matrices]]

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