Range Of A Matrix
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In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
, the column space (also called the range or image) of a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
''A'' is the
span Span may refer to: Science, technology and engineering * Span (unit), the width of a human hand * Span (engineering), a section between two intermediate supports * Wingspan, the distance between the wingtips of a bird or aircraft * Sorbitan ester ...
(set of all possible linear combinations) of its
column vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
s. The column space of a matrix is the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
or
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
of the corresponding
matrix transformation In linear algebra, linear transformations can be represented by matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fiction ...
. Let \mathbb be a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. The column space of an matrix with components from \mathbb is a
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
of the ''m''-space \mathbb^m. The
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of the column space is called the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...
of the matrix and is at most .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. A definition for matrices over a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
\mathbb is also possible. The row space is defined similarly. The row space and the column space of a matrix are sometimes denoted as and respectively. This article considers matrices of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. The row and column spaces are subspaces of the real spaces \R^n and \R^m respectively.


Overview

Let be an -by- matrix. Then # , # = number of pivots in any echelon form of , # = the maximum number of linearly independent rows or columns of . If one considers the matrix as a
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
from \mathbb^n to \mathbb^m, then the column space of the matrix equals the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of this linear transformation. The column space of a matrix is the set of all linear combinations of the columns in . If , then . The concept of row space generalizes to matrices over the field of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, or over any
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. Intuitively, given a matrix , the action of the matrix on a vector will return a linear combination of the columns of weighted by the coordinates of as coefficients. Another way to look at this is that it will (1) first project into the row space of , (2) perform an invertible transformation, and (3) place the resulting vector in the column space of . Thus the result must reside in the column space of . See
singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is related ...
for more details on this second interpretation.


Example

Given a matrix : : J = \begin 2 & 4 & 1 & 3 & 2\\ -1 & -2 & 1 & 0 & 5\\ 1 & 6 & 2 & 2 & 2\\ 3 & 6 & 2 & 5 & 1 \end the rows are \mathbf_1 = \begin 2 & 4 & 1 & 3 & 2 \end, \mathbf_2 = \begin -1 & -2 & 1 & 0 & 5 \end, \mathbf_3 = \begin 1 & 6 & 2 & 2 & 2 \end, \mathbf_4 = \begin 3 & 6 & 2 & 5 & 1 \end. Consequently, the row space of is the subspace of \R^5 spanned by . Since these four row vectors are
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
, the row space is 4-dimensional. Moreover, in this case it can be seen that they are all
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
to the vector , so it can be deduced that the row space consists of all vectors in \R^5 that are orthogonal to .


Column space


Definition

Let be a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
of
scalars Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
. Let be an matrix, with column vectors . A linear combination of these vectors is any vector of the form :c_1 \mathbf_1 + c_2 \mathbf_2 + \cdots + c_n \mathbf_n, where are scalars. The set of all possible linear combinations of is called the column space of . That is, the column space of is the
span Span may refer to: Science, technology and engineering * Span (unit), the width of a human hand * Span (engineering), a section between two intermediate supports * Wingspan, the distance between the wingtips of a bird or aircraft * Sorbitan ester ...
of the vectors . Any linear combination of the column vectors of a matrix can be written as the product of with a column vector: :\begin A \begin c_1 \\ \vdots \\ c_n \end & = & \begin a_ & \cdots & a_ \\ \vdots & \ddots & \vdots \\ a_ & \cdots & a_ \end \begin c_1 \\ \vdots \\ c_n \end = \begin c_1 a_ + \cdots + c_ a_ \\ \vdots \\ c_ a_ + \cdots + c_ a_ \end = c_1 \begin a_ \\ \vdots \\ a_ \end + \cdots + c_n \begin a_ \\ \vdots \\ a_ \end \\ & = & c_1 \mathbf_1 + \cdots + c_n \mathbf_n \end Therefore, the column space of consists of all possible products , for . This is the same as the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
(or
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
) of the corresponding
matrix transformation In linear algebra, linear transformations can be represented by matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fiction ...
.


Example

If A = \begin 1 & 0 \\ 0 & 1 \\ 2 & 0 \end, then the column vectors are and . A linear combination of v1 and v2 is any vector of the form c_1 \begin 1 \\ 0 \\ 2 \end + c_2 \begin 0 \\ 1 \\ 0 \end = \begin c_1 \\ c_2 \\ 2c_1 \end The set of all such vectors is the column space of . In this case, the column space is precisely the set of vectors satisfying the equation (using
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
, this set is a
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
through the origin in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
).


Basis

The columns of span the column space, but they may not form a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
if the column vectors are not
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
. Fortunately,
elementary row operations In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multi ...
do not affect the dependence relations between the column vectors. This makes it possible to use
row reduction In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
to find a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
for the column space. For example, consider the matrix :A = \begin 1 & 3 & 1 & 4 \\ 2 & 7 & 3 & 9 \\ 1 & 5 & 3 & 1 \\ 1 & 2 & 0 & 8 \end. The columns of this matrix span the column space, but they may not be
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
, in which case some subset of them will form a basis. To find this basis, we reduce to
reduced row echelon form In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and column echelon form means that Gaussian el ...
: :\begin 1 & 3 & 1 & 4 \\ 2 & 7 & 3 & 9 \\ 1 & 5 & 3 & 1 \\ 1 & 2 & 0 & 8 \end \sim \begin 1 & 3 & 1 & 4 \\ 0 & 1 & 1 & 1 \\ 0 & 2 & 2 & -3 \\ 0 & -1 & -1 & 4 \end \sim \begin 1 & 0 & -2 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & -5 \\ 0 & 0 & 0 & 5 \end \sim \begin 1 & 0 & -2 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end. At this point, it is clear that the first, second, and fourth columns are linearly independent, while the third column is a linear combination of the first two. (Specifically, .) Therefore, the first, second, and fourth columns of the original matrix are a basis for the column space: :\begin 1 \\ 2 \\ 1 \\ 1\end,\;\; \begin 3 \\ 7 \\ 5 \\ 2\end,\;\; \begin 4 \\ 9 \\ 1 \\ 8\end. Note that the independent columns of the reduced row echelon form are precisely the columns with pivots. This makes it possible to determine which columns are linearly independent by reducing only to
echelon form In linear algebra, a Matrix (mathematics), matrix is in echelon form if it has the shape resulting from a Gaussian elimination. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and column echelon form ...
. The above algorithm can be used in general to find the dependence relations between any set of vectors, and to pick out a basis from any spanning set. Also finding a basis for the column space of is equivalent to finding a basis for the row space of the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
matrix . To find the basis in a practical setting (e.g., for large matrices), the
singular-value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is relate ...
is typically used.


Dimension

The
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of the column space is called the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...
of the matrix. The rank is equal to the number of pivots in the
reduced row echelon form In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and column echelon form means that Gaussian el ...
, and is the maximum number of linearly independent columns that can be chosen from the matrix. For example, the 4 × 4 matrix in the example above has rank three. Because the column space is the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of the corresponding
matrix transformation In linear algebra, linear transformations can be represented by matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fiction ...
, the rank of a matrix is the same as the dimension of the image. For example, the transformation \R^4 \to \R^4 described by the matrix above maps all of \R^4 to some three-dimensional subspace. The nullity of a matrix is the dimension of the
null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel ...
, and is equal to the number of columns in the reduced row echelon form that do not have pivots. The rank and nullity of a matrix with columns are related by the equation: :\operatorname(A) + \operatorname(A) = n.\, This is known as the
rank–nullity theorem The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Theor ...
.


Relation to the left null space

The
left null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kern ...
of is the set of all vectors such that . It is the same as the
null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel ...
of the
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
of . The product of the matrix and the vector can be written in terms of the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of vectors: :A^\mathsf\mathbf = \begin \mathbf_1 \cdot \mathbf \\ \mathbf_2 \cdot \mathbf \\ \vdots \\ \mathbf_n \cdot \mathbf \end, because
row vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
s of are transposes of column vectors of . Thus if and only if is
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
(perpendicular) to each of the column vectors of . It follows that the left null space (the null space of ) is the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
to the column space of . For a matrix , the column space, row space, null space, and left null space are sometimes referred to as the ''four fundamental subspaces''.


For matrices over a ring

Similarly the column space (sometimes disambiguated as ''right'' column space) can be defined for matrices over a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
as :\sum\limits_^n \mathbf_k c_k for any , with replacement of the vector -space with "
right Rights are law, legal, social, or ethics, ethical principles of Liberty, freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convent ...
free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
", which changes the order of
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector by ...
of the vector to the scalar such that it is written in an unusual order ''vector''–''scalar''.Important only if is not
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
. Actually, this form is merely a
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of the matrix to the column vector from where the order of factors is ''preserved'', unlike the formula above.


Row space


Definition

Let be a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
of
scalars Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
. Let be an matrix, with row vectors . A linear combination of these vectors is any vector of the form :c_1 \mathbf_1 + c_2 \mathbf_2 + \cdots + c_m \mathbf_m, where are scalars. The set of all possible linear combinations of is called the row space of . That is, the row space of is the
span Span may refer to: Science, technology and engineering * Span (unit), the width of a human hand * Span (engineering), a section between two intermediate supports * Wingspan, the distance between the wingtips of a bird or aircraft * Sorbitan ester ...
of the vectors . For example, if :A = \begin 1 & 0 & 2 \\ 0 & 1 & 0 \end, then the row vectors are and . A linear combination of and is any vector of the form :c_1 \begin1 & 0 & 2\end + c_2 \begin0 & 1 & 0\end = \beginc_1 & c_2 & 2c_1\end. The set of all such vectors is the row space of . In this case, the row space is precisely the set of vectors satisfying the equation (using
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
, this set is a
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
through the origin in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
). For a matrix that represents a homogeneous
system of linear equations In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three ...
, the row space consists of all linear equations that follow from those in the system. The column space of is equal to the row space of .


Basis

The row space is not affected by
elementary row operations In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multi ...
. This makes it possible to use
row reduction In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
to find a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
for the row space. For example, consider the matrix :A = \begin 1 & 3 & 2 \\ 2 & 7 & 4 \\ 1 & 5 & 2\end. The rows of this matrix span the row space, but they may not be
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
, in which case the rows will not be a basis. To find a basis, we reduce to
row echelon form In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and column echelon form means that Gaussian ...
: , , represents the rows. : \begin \begin 1 & 3 & 2 \\ 2 & 7 & 4 \\ 1 & 5 & 2\end &\xrightarrow \begin 1 & 3 & 2 \\ 0 & 1 & 0 \\ 1 & 5 & 2\end \xrightarrow \begin 1 & 3 & 2 \\ 0 & 1 & 0 \\ 0 & 2 & 0\end \\ &\xrightarrow \begin 1 & 3 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end \xrightarrow \begin 1 & 0 & 2 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end. \end Once the matrix is in echelon form, the nonzero rows are a basis for the row space. In this case, the basis is . Another possible basis comes from a further reduction.The example is valid over the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s, and other
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
s. It is not necessarily correct over fields and rings with non-zero characteristic.
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 In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and column echelon form means that Gaussian el ...
, then the resulting basis is uniquely determined by the row space. It is sometimes convenient to find a basis for the row space from among the rows of the original matrix instead (for example, this result is useful in giving an elementary proof that the determinantal rank of a matrix is equal to its rank). Since row operations can affect linear dependence relations of the row vectors, such a basis is instead found indirectly using the fact that the column space of is equal to the row space of . Using the example matrix above, find and reduce it to row echelon form: : A^ = \begin 1 & 2 & 1 \\ 3 & 7 & 5 \\ 2 & 4 & 2\end \sim \begin 1 & 2 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0\end. The pivots indicate that the first two columns of form a basis of the column space of . Therefore, the first two rows of (before any row reductions) also form a basis of the row space of .


Dimension

The
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of the row space is called the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...
of the matrix. This is the same as the maximum number of linearly independent rows that can be chosen from the matrix, or equivalently the number of pivots. For example, the 3 × 3 matrix in the example above has rank two. The rank of a matrix is also equal to the dimension of the
column space In linear algebra, the column space (also called the range or image) of a matrix ''A'' is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix ...
. The dimension of the
null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel ...
is called the nullity of the matrix, and is related to the rank by the following equation: :\operatorname(A) + \operatorname(A) = n, where is the number of columns of the matrix . The equation above is known as the
rank–nullity theorem The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Theor ...
.


Relation to the null space

The
null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel ...
of matrix is the set of all vectors for which . The product of the matrix and the vector can be written in terms of the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of vectors: :A\mathbf = \begin \mathbf_1 \cdot \mathbf \\ \mathbf_2 \cdot \mathbf \\ \vdots \\ \mathbf_m \cdot \mathbf \end, where are the row vectors of . Thus if and only if is
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
(perpendicular) to each of the row vectors of . It follows that the null space of is the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
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 The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Theor ...
(see
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
above). The row space and null space are two of the
four fundamental subspaces In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the Domain of a function, domain of the map which is mapped to the zero vector. That is, given a linear map between two vector space ...
associated with a matrix (the other two being the
column space In linear algebra, the column space (also called the range or image) of a matrix ''A'' is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix ...
and
left null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kern ...
).


Relation to coimage

If and are
vector spaces In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
, then the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of a
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
is the set of vectors for which . The kernel of a linear transformation is analogous to the null space of a matrix. If is an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
, then the orthogonal complement to the kernel can be thought of as a generalization of the row space. This is sometimes called the
coimage In algebra, the coimage of a homomorphism :f : A \rightarrow B is the quotient :\text f = A/\ker(f) of the domain by the kernel. The coimage is canonically isomorphic to the image by the first isomorphism theorem, when that theorem applies. M ...
of . The transformation is one-to-one on its coimage, and the coimage maps isomorphically onto the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of . When is not an inner product space, the coimage of can be defined as the quotient space .


See also

*
Euclidean subspace In geometry, a flat or Euclidean subspace is a subset of a Euclidean space that is itself a Euclidean space (of lower dimension). The flats in two-dimensional space are points and lines, and the flats in three-dimensional space are points, lin ...


References & Notes


Further reading

* * * * * * * * *


External links

* * *
MIT Linear Algebra Lecture on the Four Fundamental Subspaces
at Google Video, from
MIT OpenCourseWare MIT OpenCourseWare (MIT OCW) is an initiative of the Massachusetts Institute of Technology (MIT) to publish all of the educational materials from its Post-secondary education, undergraduate- and Quaternary education, graduate-level courses online ...

Khan Academy video tutorialLecture on column space and nullspace by Gilbert Strang of MIT
{{linear algebra Abstract algebra Linear algebra Matrices