In

^{3}.
Take ''W'' to be the set of all vectors in ''V'' whose last component is 0.
Then ''W'' is a subspace of ''V''.
''Proof:''
#Given u and v in ''W'', then they can be expressed as and . Then . Thus, u + v is an element of ''W'', too.
#Given u in ''W'' and a scalar ''c'' in R, if again, then . Thus, ''c''u is an element of ''W'' too.

^{2}.
Take ''W'' to be the set of points (''x'', ''y'') of R^{2} such that ''x'' = ''y''.
Then ''W'' is a subspace of R^{2}.
''Proof:''
#Let and be elements of ''W'', that is, points in the plane such that ''p''_{1} = ''p''_{2} and ''q''_{1} = ''q''_{2}. Then ; since ''p''_{1} = ''p''_{2} and ''q''_{1} = ''q''_{2}, then ''p''_{1} + ''q''_{1} = ''p''_{2} + ''q''_{2}, so p + q is an element of ''W''.
#Let p = (''p''_{1}, ''p''_{2}) be an element of ''W'', that is, a point in the plane such that ''p''_{1} = ''p''_{2}, and let ''c'' be a scalar in R. Then ; since ''p''_{1} = ''p''_{2}, then ''cp''_{1} = ''cp''_{2}, so ''c''p is an element of ''W''.
In general, any subset of the real coordinate space R^{''n''} that is defined by a system of homogeneous

^{R} of all ^{R}.
''Proof:''
#We know from calculus that .
#We know from calculus that the sum of continuous functions is continuous.
#Again, we know from calculus that the product of a continuous function and a number is continuous.

^{''n''}:
:$\backslash left\backslash .$
For example, the set of all vectors (''x'', ''y'', ''z'') (over real or ^{''k''} will be the dimension of the

^{''n''} can be described as the null space of some matrix (see below for more).

^{''n''} described by a system of homogeneous linear ^{3}, if ''K'' is a

_{1}, v_{2}, ... , v_{''k''} is any vector of the form
:$t\_1\; \backslash mathbf\_1\; +\; \backslash cdots\; +\; t\_k\; \backslash mathbf\_k.$
The set of all possible linear combinations is called the span:
:$\backslash text\; \backslash \; =\; \backslash left\backslash \; .$
If the vectors v_{1}, ... , v_{''k''} have ''n'' components, then their span is a subspace of ''K''^{''n''}. Geometrically, the span is the flat through the origin in ''n''-dimensional space determined by the points v_{1}, ... , v_{''k''}.
; Example
: The ''xz''-plane in R^{3} can be parameterized by the equations
::$x\; =\; t\_1,\; \backslash ;\backslash ;\backslash ;\; y\; =\; 0,\; \backslash ;\backslash ;\backslash ;\; z\; =\; t\_2.$
:As a subspace, the ''xz''-plane is spanned by the vectors (1, 0, 0) and (0, 0, 1). Every vector in the ''xz''-plane can be written as a linear combination of these two:
::$(t\_1,\; 0,\; t\_2)\; =\; t\_1(1,0,0)\; +\; t\_2(0,0,1)\backslash text$
:Geometrically, this corresponds to the fact that every point on the ''xz''-plane can be reached from the origin by first moving some distance in the direction of (1, 0, 0) and then moving some distance in the direction of (0, 0, 1).

^{''n''} spanned by the column vectors of ''A''.
The row space of a matrix is the subspace spanned by its row vectors. The row space is interesting because it is the

^{''n''} determined by ''k'' parameters (or spanned by ''k'' vectors) has dimension ''k''. However, there are exceptions to this rule. For example, the subspace of ''K''^{3} spanned by the three vectors (1, 0, 0), (0, 0, 1), and (2, 0, 3) is just the ''xz''-plane, with each point on the plane described by infinitely many different values of .
In general, vectors v_{1}, ... , v_{''k''} are called linearly independent if
:$t\_1\; \backslash mathbf\_1\; +\; \backslash cdots\; +\; t\_k\; \backslash mathbf\_k\; \backslash ;\backslash ne\backslash ;\; u\_1\; \backslash mathbf\_1\; +\; \backslash cdots\; +\; u\_k\; \backslash mathbf\_k$
for
(''t''_{1}, ''t''_{2}, ... , ''t_{k}'') ≠ (''u''_{1}, ''u''_{2}, ... , ''u_{k}'').This definition is often stated differently: vectors v_{1}, ..., v_{''k''} are linearly independent if
for . The two definitions are equivalent.
If are linearly independent, then the coordinates for a vector in the span are uniquely determined.
A basis for a subspace ''S'' is a set of linearly independent vectors whose span is ''S''. The number of elements in a basis is always equal to the geometric dimension of the subspace. Any spanning set for a subspace can be changed into a basis by removing redundant vectors (see § Algorithms below for more).
; Example
: Let ''S'' be the subspace of R^{4} defined by the equations
::$x\_1\; =\; 2\; x\_2\backslash ;\backslash ;\backslash ;\backslash ;\backslash text\backslash ;\backslash ;\backslash ;\backslash ;x\_3\; =\; 5x\_4.$
:Then the vectors (2, 1, 0, 0) and (0, 0, 5, 1) are a basis for ''S''. In particular, every vector that satisfies the above equations can be written uniquely as a linear combination of the two basis vectors:
::$(2t\_1,\; t\_1,\; 5t\_2,\; t\_2)\; =\; t\_1(2,\; 1,\; 0,\; 0)\; +\; t\_2(0,\; 0,\; 5,\; 1).$
:The subspace ''S'' is two-dimensional. Geometrically, it is the plane in R^{4} passing through the points (0, 0, 0, 0), (2, 1, 0, 0), and (0, 0, 5, 1).

^{''n''} are equal.

^{''n''}, and a vector v with ''n'' components.
:Output Determines whether v is an element of ''S''
:# Create a (''k'' + 1) × ''n'' matrix ''A'' whose rows are the vectors b_{1}, ... , b_{''k''} and v.
:# Use elementary row operations to put ''A'' into row echelon form.
:# If the echelon form has a row of zeroes, then the vectors are linearly dependent, and therefore .

^{''n''}, and a vector
:Output Numbers ''t''_{1}, ''t''_{2}, ..., ''t''_{''k''} such that
:# Create an _{1},...,b_{''k''} , with the last column being v.
:# Use elementary row operations to put ''A'' into reduced row echelon form.
:# Express the final column of the reduced echelon form as a linear combination of the first ''k'' columns. The coefficients used are the desired numbers . (These should be precisely the first ''k'' entries in the final column of the reduced echelon form.)
If the final column of the reduced row echelon form contains a pivot, then the input vector v does not lie in ''S''.

_{i}'', choose a vector in the null space for which and the remaining free variables are zero. The resulting collection of vectors is a basis for the null space of ''A''.
See the article on null space for an

^{''n''}
:Output An (''n'' − ''k'') × ''n'' matrix whose null space is ''S''.
:# Create a matrix ''A'' whose rows are .
:# Use elementary row operations to put ''A'' into reduced row echelon form.
:# Let be the columns of the reduced row echelon form. For each column without a pivot, write an equation expressing the column as a linear combination of the columns with pivots.
:# This results in a homogeneous system of ''n'' − ''k'' linear equations involving the variables c_{1},...,c_{''n''}. The matrix corresponding to this system is the desired matrix with nullspace ''S''.
; Example
:If the reduced row echelon form of ''A'' is
::$\backslash left;\; href="/html/ALL/s/\backslash begin\; 1\_\_0\_\_-3\_\_0\_\_\_2\_\_0\_\backslash \backslash \; 0\_\_1\_\_\_5\_\_0\_\_-1\_\_4\_\backslash \backslash \; 0\_\_0\_\_\_0\_\_1\_\_\_7\_\_-9\_\backslash \backslash \; 0\_\_\backslash ;\backslash ;\backslash ;\backslash ;\backslash ;0\_\_\_\backslash ;\backslash ;\backslash ;\backslash ;\backslash ;0\_\_\backslash ;\backslash ;\backslash ;\backslash ;\backslash ;0\_\_\_\backslash ;\backslash ;\backslash ;\backslash ;\backslash ;0\_\_\backslash ;\backslash ;\backslash ;\backslash ;\backslash ;0\_\backslash end\_\backslash ,\backslash right.html"\; ;"title="\backslash begin\; 1\; 0\; -3\; 0\; 2\; 0\; \backslash \backslash \; 0\; 1\; 5\; 0\; -1\; 4\; \backslash \backslash \; 0\; 0\; 0\; 1\; 7\; -9\; \backslash \backslash \; 0\; \backslash ;\backslash ;\backslash ;\backslash ;\backslash ;0\; \backslash ;\backslash ;\backslash ;\backslash ;\backslash ;0\; \backslash ;\backslash ;\backslash ;\backslash ;\backslash ;0\; \backslash ;\backslash ;\backslash ;\backslash ;\backslash ;0\; \backslash ;\backslash ;\backslash ;\backslash ;\backslash ;0\; \backslash end\; \backslash ,\backslash right">\backslash begin\; 1\; 0\; -3\; 0\; 2\; 0\; \backslash \backslash \; 0\; 1\; 5\; 0\; -1\; 4\; \backslash \backslash \; 0\; 0\; 0\; 1\; 7\; -9\; \backslash \backslash \; 0\; \backslash ;\backslash ;\backslash ;\backslash ;\backslash ;0\; \backslash ;\backslash ;\backslash ;\backslash ;\backslash ;0\; \backslash ;\backslash ;\backslash ;\backslash ;\backslash ;0\; \backslash ;\backslash ;\backslash ;\backslash ;\backslash ;0\; \backslash ;\backslash ;\backslash ;\backslash ;\backslash ;0\; \backslash end\; \backslash ,\backslash right$
:then the column vectors satisfy the equations
::$\backslash begin\; \backslash mathbf\_3\; \&=\; -3\backslash mathbf\_1\; +\; 5\backslash mathbf\_2\; \backslash \backslash \; \backslash mathbf\_5\; \&=\; 2\backslash mathbf\_1\; -\; \backslash mathbf\_2\; +\; 7\backslash mathbf\_4\; \backslash \backslash \; \backslash mathbf\_6\; \&=\; 4\backslash mathbf\_2\; -\; 9\backslash mathbf\_4\; \backslash end$
:It follows that the row vectors of ''A'' satisfy the equations
::$\backslash begin\; x\_3\; \&=\; -3x\_1\; +\; 5x\_2\; \backslash \backslash \; x\_5\; \&=\; 2x\_1\; -\; x\_2\; +\; 7x\_4\; \backslash \backslash \; x\_6\; \&=\; 4x\_2\; -\; 9x\_4.\; \backslash end$
:In particular, the row vectors of ''A'' are a basis for the null space of the corresponding matrix.

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, and more specifically 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 matrix (mat ...

, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats
Flat or flats may refer to:
Architecture
* Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries
Arts and entertainment
* Flat (music), a symbol () which denotes a lower pitch
* Flat (soldier), a ...

and affine subspace
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s. In the case of vector spaces over the reals, linear subspaces, flats, and affine subspaces are also called ''linear manifolds'' for emphasizing that there are also manifold
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s. is a vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

that is a subset
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of some larger vector space. A linear subspace is usually simply called a ''subspace'' when the context serves to distinguish it from other types of subspaces.
Definition

If ''V'' is a vector space over afield
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 grassl ...

''K'' and if ''W'' is a subset of ''V'', then ''W'' is a linear subspace of ''V'' if under the operations of ''V'', ''W'' is a vector space over ''K''. Equivalently, a nonempty #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

subset ''W'' is a subspace of ''V'' if, whenever are elements of ''W'' and are elements of ''K'', it follows that is in ''W''.
As a corollary, all vector spaces are equipped with at least two (possibly different) linear subspaces: the zero vector space
Morphisms to and from the zero object
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geomet ...

consisting of the zero vector
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

alone and the entire vector space itself. These are called the trivial subspaces of the vector space.
Examples

Example I

Let the field ''K'' be the set R ofreal number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s, and let the vector space ''V'' be the real coordinate space
In mathematics, a real coordinate space of dimension , written ( ) or is a coordinate space over the real numbers. This means that it is the set of the tuple, -tuples of real numbers (sequences of real numbers). With component-wise addition a ...

RExample II

Let the field be R again, but now let the vector space ''V'' be theCartesian plane
A Cartesian coordinate system (, ) in a plane
Plane or planes may refer to:
* Airplane
An airplane or aeroplane (informally plane) is a fixed-wing aircraft
A fixed-wing aircraft is a heavier-than-air flying machine
Early fly ...

Rlinear equation
In mathematics, a linear equation is an equation that may be put in the form
:a_1x_1+\cdots +a_nx_n+b=0,
where x_1, \ldots, x_n are the variable (mathematics), variables (or unknown (mathematics), unknowns), and b, a_1, \ldots, a_n are the coeffi ...

s will yield a subspace.
(The equation in example I was ''z'' = 0, and the equation in example II was ''x'' = ''y''.)
Geometrically, these subspaces are points, lines, planes and spaces that pass through the point 0.
Example III

Again take the field to be R, but now let the vector space ''V'' be the set Rfunction
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

s from R to R.
Let C(R) be the subset consisting of continuous function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s.
Then C(R) is a subspace of RExample IV

Keep the same field and vector space as before, but now consider the set Diff(R) of alldifferentiable function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s.
The same sort of argument as before shows that this is a subspace too.
Examples that extend these themes are common in functional analysis
Functional analysis is a branch of mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...

.
Properties of subspaces

From the definition of vector spaces, it follows that subspaces are nonempty, and are closed under sums and under scalar multiples. Equivalently, subspaces can be characterized by the property of being closed under linear combinations. That is, a nonempty set ''W'' is a subspaceif and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...

every linear combination of finite
Finite is the opposite of infinite
Infinite may refer to:
Mathematics
*Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (band), a South Korean boy band
*''Infin ...

ly many elements of ''W'' also belongs to ''W''.
The equivalent definition states that it is also equivalent to consider linear combinations of two elements at a time.
In a topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space (an Abstra ...

''X'', a subspace ''W'' need not be topologically closed, but a finite-dimensional
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

subspace is always closed. The same is true for subspaces of finite codimension
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

(i.e., subspaces determined by a finite number of continuous linear functional
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s).
Descriptions

Descriptions of subspaces include the solution set to a homogeneoussystem of linear equations
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, the subset of Euclidean space described by a system of homogeneous linear parametric equations
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, the span of a collection of vectors, and the null space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, column space
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 a ...

, and row space
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 a ...

of a matrix
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols, or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the material in between a eukaryot ...

. Geometrically (especially over the field of real numbers and its subfields), a subspace is a flat
Flat or flats may refer to:
Architecture
* Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries
Arts and entertainment
* Flat (music), a symbol () which denotes a lower pitch
* Flat (soldier), a ...

in an ''n''-space that passes through the origin.
A natural description of a 1-subspace is the scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module (mathematics), module in abstract algebra). In common geometrical contexts, scalar multiplication of a re ...

of one non-zero
0 (zero) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

vector v to all possible scalar values. 1-subspaces specified by two vectors are equal if and only if one vector can be obtained from another with scalar multiplication:
:$\backslash exist\; c\backslash in\; K:\; \backslash mathbf\text{'}\; =\; c\backslash mathbf\backslash text\backslash mathbf\; =\; \backslash frac\backslash mathbf\text{'}\backslash text$
This idea is generalized for higher dimensions with linear span
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

, but criteria for equality
Equality may refer to:
Society
* Political equality, in which all members of a society are of equal standing
** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elite ...

of ''k''-spaces specified by sets of ''k'' vectors are not so simple.
A dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
** . . . see more cases in :Duality theories
* Dual ...

description is provided with linear functionals
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field (mathematics), field of scalar (mathematics), scalars (often, the real numbers or the complex numbers).
...

(usually implemented as linear equations). One non-zero
0 (zero) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

linear functional F specifies its 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 ...

subspace F = 0 of codimension 1. Subspaces of codimension 1 specified by two linear functionals are equal, if and only if one functional can be obtained from another with scalar multiplication (in the dual space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

):
:$\backslash exist\; c\backslash in\; K:\; \backslash mathbf\text{'}\; =\; c\backslash mathbf\backslash text\backslash mathbf\; =\; \backslash frac\backslash mathbf\text{'}\backslash text$
It is generalized for higher codimensions with a system of equations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

. The following two subsections will present this latter description in details, and the remaining four subsections further describe the idea of linear span.
Systems of linear equations

The solution set to any homogeneoussystem of linear equations
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

with ''n'' variables is a subspace in the coordinate space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

''K''rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

s) satisfying the equations
:$x\; +\; 3y\; +\; 2z\; =\; 0\; \backslash ;\backslash ;\backslash ;\backslash ;\backslash text\backslash ;\backslash ;\backslash ;\backslash ;\; 2x\; -\; 4y\; +\; 5z\; =\; 0$
is a one-dimensional subspace. More generally, that is to say that given a set of ''n'' independent functions, the dimension of the subspace in ''K''null set
In mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...

of ''A'', the composite matrix of the ''n'' functions.
Null space of a matrix

In a finite-dimensional space, a homogeneous system of linear equations can be written as a single matrix equation: :$A\backslash mathbf\; =\; \backslash mathbf.$ The set of solutions to this equation is known as thenull space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of the matrix. For example, the subspace described above is the null space of the matrix
:$A\; =\; \backslash begin\; 1\; \&\; 3\; \&\; 2\; \backslash \backslash \; 2\; \&\; -4\; \&\; 5\; \backslash end\; .$
Every subspace of ''K''Linear parametric equations

The subset of ''K''parametric equations
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

is a subspace:
:$\backslash left\backslash .$
For example, the set of all vectors (''x'', ''y'', ''z'') parameterized by the equations
:$x\; =\; 2t\_1\; +\; 3t\_2,\backslash ;\backslash ;\backslash ;\backslash ;y\; =\; 5t\_1\; -\; 4t\_2,\backslash ;\backslash ;\backslash ;\backslash ;\backslash text\backslash ;\backslash ;\backslash ;\backslash ;z\; =\; -t\_1\; +\; 2t\_2$
is a two-dimensional subspace of ''K''number field
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

(such as real or rational numbers).Generally, ''K'' can be any field of such characteristic
Characteristic (from the Greek word for a property, attribute or trait
Trait may refer to:
* Phenotypic trait in biology, which involve genes and characteristics of organisms
* Trait (computer programming), a model for structuring object-oriented ...

that the given integer matrix has the appropriate rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking
A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...

in it. All fields include integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s, but some integers may equal to zero in some fields.
Span of vectors

In linear algebra, the system of parametric equations can be written as a single vector equation: :$\backslash begin\; x\; \backslash \backslash \; y\; \backslash \backslash \; z\; \backslash end\; \backslash ;=\backslash ;\; t\_1\; \backslash !\backslash begin\; 2\; \backslash \backslash \; 5\; \backslash \backslash \; -1\; \backslash end\; +\; t\_2\; \backslash !\backslash begin\; 3\; \backslash \backslash \; -4\; \backslash \backslash \; 2\; \backslash end.$ The expression on the right is called a linear combination of the vectors (2, 5, −1) and (3, −4, 2). These two vectors are said to span the resulting subspace. In general, a linear combination of vectors vColumn space and row space

A system of linear parametric equations in a finite-dimensional space can also be written as a single matrix equation: :$\backslash mathbf\; =\; A\backslash mathbf\backslash ;\backslash ;\backslash ;\backslash ;\backslash text\backslash ;\backslash ;\backslash ;\backslash ;A\; =\; \backslash left;\; href="/html/ALL/s/\backslash begin\_2\_\_3\_\_\backslash \backslash \_5\_\_\backslash ;\backslash ;-4\_\_\backslash \backslash \_-1\_\_2\_\_\backslash end\_\backslash ,\backslash right.html"\; ;"title="\backslash begin\; 2\; 3\; \backslash \backslash \; 5\; \backslash ;\backslash ;-4\; \backslash \backslash \; -1\; 2\; \backslash end\; \backslash ,\backslash right">\backslash begin\; 2\; 3\; \backslash \backslash \; 5\; \backslash ;\backslash ;-4\; \backslash \backslash \; -1\; 2\; \backslash end\; \backslash ,\backslash right$ In this case, the subspace consists of all possible values of the vector x. In linear algebra, this subspace is known as the column space (orimage
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...

) of the matrix ''A''. It is precisely the subspace of ''K''orthogonal complement In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of the null space (see below).
Independence, basis, and dimension

In general, a subspace of ''K''Operations and relations on subspaces

Inclusion

The set-theoretical inclusion binary relation specifies apartial order
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

on the set of all subspaces (of any dimension).
A subspace cannot lie in any subspace of lesser dimension. If dim ''U'' = ''k'', a finite number, and ''U'' ⊂ ''W'', then dim ''W'' = ''k'' if and only if ''U'' = ''W''.
Intersection

Given subspaces ''U'' and ''W'' of a vector space ''V'', then theirintersection
In mathematics
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''U'' ∩ ''W'' := is also a subspace of ''V''.
''Proof:''
# Let v and w be elements of ''U'' ∩ ''W''. Then v and w belong to both ''U'' and ''W''. Because ''U'' is a subspace, then v + w belongs to ''U''. Similarly, since ''W'' is a subspace, then v + w belongs to ''W''. Thus, v + w belongs to ''U'' ∩ ''W''.
# Let v belong to ''U'' ∩ ''W'', and let ''c'' be a scalar. Then v belongs to both ''U'' and ''W''. Since ''U'' and ''W'' are subspaces, ''c''v belongs to both ''U'' and ''W''.
# Since ''U'' and ''W'' are vector spaces, then 0 belongs to both sets. Thus, 0 belongs to ''U'' ∩ ''W''.
For every vector space ''V'', the set and ''V'' itself are subspaces of ''V''.
Sum

If ''U'' and ''W'' are subspaces, their sum is the subspace : $U\; +\; W\; =\; \backslash left\backslash .$Vector space related operators. For example, the sum of two lines is the plane that contains them both. The dimension of the sum satisfies the inequality : $\backslash max(\backslash dim\; U,\backslash dim\; W)\; \backslash leq\; \backslash dim(U\; +\; W)\; \backslash leq\; \backslash dim(U)\; +\; \backslash dim(W).$ Here, the minimum only occurs if one subspace is contained in the other, while the maximum is the most general case. The dimension of the intersection and the sum are related by the following equation: : $\backslash dim(U+W)\; =\; \backslash dim(U)\; +\; \backslash dim(W)\; -\; \backslash dim(U\; \backslash cap\; W).$ A set of subspaces is independent when the only intersection between any pair of subspaces is the trivial subspace. Thedirect sum
The direct sum is an operation from abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...

is the sum of independent subspaces, written as $U\; \backslash oplus\; W$. An equivalent restatement is that a direct sum is a subspace sum under the condition that every subspace contributes to the span of the sum.
The dimension of a direct sum $U\; \backslash oplus\; W$ is the same as the sum of subspaces, but may be shortened because the dimension of the trivial subspace is zero.
$$\backslash dim\; (U\; \backslash oplus\; W)\; =\; \backslash dim\; (U)\; +\; \backslash dim\; (W)$$
Lattice of subspaces

The operationsintersection
In mathematics
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and sum make the set of all subspaces a bounded modular lattice
In the branch of mathematics called order theory
Order theory is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), ...

, where the subspace, the least element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined duality (order theory), dually, ...

, is an identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

of the sum operation, and the identical subspace ''V'', the greatest element, is an identity element of the intersection operation.
Orthogonal complements

If $V$ is aninner product space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

and $N$ is a subset of $V$, then the orthogonal complement In the mathematical
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of $N$, denoted $N^$, is again a subspace. If $V$ is finite-dimensional and $N$ is a subspace, then the dimensions of $N$ and $N^$ satisfy the complementary relationship $\backslash dim\; (N)\; +\; \backslash dim\; (N^)\; =\; \backslash dim\; (V)$. Moreover, no vector is orthogonal to itself, so $N\; \backslash cap\; N^\backslash perp\; =\; \backslash $ and $V$ is the direct sum
The direct sum is an operation from abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...

of $N$ and $N^$. Applying orthogonal complements twice returns the original subspace: $(N^)^\; =\; N$ for every subspace $N$. p. 195, § 6.51
This operation, understood as negation
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, ...

($\backslash neg$), makes the lattice of subspaces a (possibly infinite
Infinite may refer to:
Mathematics
*Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (band), a South Korean boy band
*''Infinite'' (EP), debut EP of American mus ...

) orthocomplemented lattice (although not a distributive lattice).
In spaces with other bilinear form
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s, some but not all of these results still hold. In pseudo-Euclidean spaceIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s and symplectic vector space In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

s, for example, orthogonal complements exist. However, these spaces may have null vector
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s that are orthogonal to themselves, and consequently there exist subspaces $N$ such that $N\; \backslash cap\; N^\; \backslash ne\; \backslash $. As a result, this operation does not turn the lattice of subspaces into a Boolean algebra (nor a Heyting algebra
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Arend Heyting (; 9 May 1898 – 9 July 1980) was a Dutch
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* Something of, from, or related to the Netherlands
* Dutch people ()
* Dutch language ()
*Dutch language , spoken in Belgium (also referred as ''fl ...

).
Algorithms

Most algorithms for dealing with subspaces involve row reduction. This is the process of applyingelementary row operation In mathematics, an elementary matrix is a matrix (mathematics), 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 multip ...

s to a matrix, until it reaches either row 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 ...

or reduced row 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 ...

. Row reduction has the following important properties:
# The reduced matrix has the same null space as the original.
# Row reduction does not change the span of the row vectors, i.e. the reduced matrix has the same row space as the original.
# Row reduction does not affect the linear dependence of the column vectors.
Basis for a row space

:Input An ''m'' × ''n'' matrix ''A''. :Output A basis for the row space of ''A''. :# Use elementary row operations to put ''A'' into row echelon form. :# The nonzero rows of the echelon form are a basis for the row space of ''A''. See the article onrow space
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 a ...

for an example
Example may refer to:
* ''exempli gratia
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{{Latin phrases
E ...'' (e.g.), usually read out in English as "for example"
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.
If we instead put the matrix ''A'' into reduced row echelon form, then the resulting basis for the row space is uniquely determined. This provides an algorithm for checking whether two row spaces are equal and, by extension, whether two subspaces of ''K''Subspace membership

:Input A basis for a subspace ''S'' of ''K''Basis for a column space

:Input An ''m'' × ''n'' matrix ''A'' :Output A basis for the column space of ''A'' :# Use elementary row operations to put ''A'' into row echelon form. :# Determine which columns of the echelon form have pivots. The corresponding columns of the original matrix are a basis for the column space. See the article on column space for anexample
Example may refer to:
* ''exempli gratia
Notes and references
Notes
References
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*
*
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{{Latin phrases
E ...'' (e.g.), usually read out in English as "for example"
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.
This produces a basis for the column space that is a subset of the original column vectors. It works because the columns with pivots are a basis for the column space of the echelon form, and row reduction does not change the linear dependence relationships between the columns.
Coordinates for a vector

:Input A basis for a subspace ''S'' of ''K''augmented matrix
Augment or augmentation may refer to:
Language
* Augment (Indo-European), a syllable added to the beginning of the word in certain Indo-European languages
* Augment (Bantu languages), a morpheme that is prefixed to the noun class prefix of nouns ...

''A'' whose columns are bBasis for a null space

:Input An ''m'' × ''n'' matrix ''A''. :Output A basis for the null space of ''A'' :# Use elementary row operations to put ''A'' in reduced row echelon form. :# Using the reduced row echelon form, determine which of the variables are free. Write equations for the dependent variables in terms of the free variables. :# For each free variable ''xexample
Example may refer to:
* ''exempli gratia
Notes and references
Notes
References
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Further reading
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{{Latin phrases
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Basis for the sum and intersection of two subspaces

Given two subspaces and of , a basis of the sum $U\; +\; W$ and the intersection $U\; \backslash cap\; W$ can be calculated using the Zassenhaus algorithmEquations for a subspace

:Input A basis for a subspace ''S'' of ''K''See also

* Cyclic subspace *Invariant subspaceIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

* Multilinear subspace learning
Multilinear subspace learning is an approach to dimensionality reduction.M. A. O. Vasilescu, D. Terzopoulos (2003"Multilinear Subspace Analysis of Image Ensembles" "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVP ...

* Quotient space (linear algebra)
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 an ...

* Signal subspace
* Subspace topology In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...

Notes

Citations

Sources

Textbook

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* *External links

* * {{Linear algebra Linear algebra Articles containing proofs Operator theory Functional analysis ru:Векторное подпространство