1 Definition and axioms
A vector space is a set equipped with two operations: vector addition and scalar multiplication. The elements of the set are called vectors, while the scalars usually come from a field such as the real numbers or complex numbers. Together, these operations must satisfy a standard list of axioms that capture the idea of linear combination and ensure that algebraic manipulation behaves consistently.
Vector spaces provide a common language for many mathematical objects that may look very different on the surface. Points in coordinate form, polynomials, functions, matrices, and sequences can all be treated as vectors when the relevant operations are defined appropriately. This abstraction makes it possible to study geometry, algebra, and analysis through a unified framework.
1.1 Vectors and scalars
In a vector space, vectors are the objects being added and rescaled. Scalars are the coefficients used to multiply vectors and form linear combinations. The scalars are drawn from a fixed field, which supplies addition, multiplication, zero, one, and multiplicative inverses for nonzero elements.
The distinction between vectors and scalars is essential. Scalars determine how vectors are stretched, reversed, or combined, but they are not themselves part of the vector space unless the space has been built that way. In many familiar settings, the scalars are real numbers or complex numbers.
1.2 Vector addition
Vector addition is a binary operation that combines two vectors to produce another vector in the same space. This operation is typically written with the symbol +. It must be closed, meaning the sum of any two vectors in the space is again a vector in the space.
Addition is intended to model the idea of combining objects of the same type. In coordinate spaces, it is performed component by component; for functions, it is done pointwise; for matrices, entrywise. The chosen operation must fit the structure of the set and satisfy the vector space axioms.
1.3 Scalar multiplication
Scalar multiplication assigns to each scalar and vector a new vector, usually written by juxtaposition or with a dot. Multiplying by a scalar changes the magnitude of a vector and, in many settings, may also reverse its direction if the scalar is negative.
Like addition, scalar multiplication must be closed. It is defined in ways that reflect the nature of the vectors involved: scaling coordinates, multiplying each matrix entry, or multiplying a function value pointwise. The axioms ensure that scaling interacts predictably with addition and with multiplication of scalars.
1.4 Vector space axioms
The axioms of a vector space specify the exact algebraic laws that addition and scalar multiplication must obey. These laws are what make linear algebra possible. They ensure that expressions involving sums, scalar multiples, and combinations of vectors can be rearranged without ambiguity.
The axioms are usually grouped into several families covering addition, identity and inverse elements, and distributive and compatibility rules. When all of them hold, the set becomes a vector space over the chosen field.
1.4.1 Associativity and commutativity of addition
Vector addition must be associative, so that the order of grouping does not matter when adding three vectors. It must also be commutative, so the order of two addends does not affect the sum. These properties allow sums of many vectors to be written unambiguously.
1.4.2 Additive identity and inverses
There must be a zero vector that acts as an additive identity: adding it to any vector leaves that vector unchanged. Every vector must also have an additive inverse, meaning another vector whose sum with it is the zero vector. These requirements make subtraction within the space possible.
1.4.3 Distributive and compatibility laws
Scalar multiplication must distribute over vector addition and over scalar addition. In addition, multiplying by a product of scalars must agree with successive scalar multiplication, and multiplying by the scalar one must leave each vector unchanged. These rules connect the field structure of the scalars with the additive structure of the vectors.
1.5 Examples of vector spaces
Common examples include coordinate spaces, spaces of polynomials, and spaces of functions. Less familiar examples arise from matrices of fixed size, sequences with suitable coordinatewise operations, and solution sets of homogeneous linear equations. Each example demonstrates that the definition depends on the algebraic laws, not on any particular geometric picture.
2 Basic properties
Once the axioms are established, many useful facts follow automatically. These properties are often used without comment in calculations, because they are consequences of the fundamental rules rather than additional assumptions.
2.1 Uniqueness of identity and inverses
The zero vector is unique, since two additive identities would have to equal each other by the identity property. Likewise, each vector has exactly one additive inverse. These uniqueness results confirm that the additive structure of a vector space is well defined.
2.2 Cancellation laws
If a vector is added to both sides of an equation, it can be canceled from the equation. This is a direct consequence of additive inverses and associativity. Cancellation is one of the most common tools in vector space calculations.
2.3 Zero vector and zero scalar properties
Multiplying any vector by the scalar zero gives the zero vector. Multiplying the zero vector by any scalar also produces the zero vector. These facts are useful for simplifying expressions and follow from the distributive laws.
2.4 Immediate consequences of the axioms
Many familiar algebraic identities can be derived from the axioms. For example, multiplying a vector by minus one produces its additive inverse, and the scalar negative distributes across vectors as expected. Such consequences help streamline proofs and computations throughout linear algebra.
3 Examples of vector spaces
Examples are central to understanding the concept of a vector space. They show that the abstraction is broad enough to include many different mathematical settings while still retaining a common structure.
3.1 Euclidean coordinate spaces
The space of ordered n-tuples of real or complex numbers is the standard example of a finite-dimensional vector space. Addition and scalar multiplication are defined component by component. These spaces form the basis for coordinate geometry and much of applied linear algebra.
3.2 Matrices
The set of all matrices of a fixed size over a field is a vector space. Matrix addition and scalar multiplication are defined entrywise. Because the dimensions must be fixed, matrices of different sizes do not belong to the same vector space under these operations.
3.3 Polynomials
Polynomials with coefficients in a field form a vector space under ordinary polynomial addition and scalar multiplication. The degree of a polynomial does not restrict its membership, so the space is infinite-dimensional. This example is important in algebra and approximation theory.
3.4 Functions
Sets of functions can form vector spaces when addition and scalar multiplication are defined pointwise. For instance, all real-valued functions on a given domain make a vector space over the real numbers. Such spaces are fundamental in analysis, where linear methods are applied to functions rather than finite coordinate lists.
3.5 Sequences
Sequences of scalars can be added and scaled term by term, producing a vector space. Depending on the class of sequences chosen, the space may be finite or infinite dimensional. Sequence spaces are important in analysis and the study of series.
3.6 Solution sets of linear homogeneous equations
The set of solutions to a homogeneous linear system forms a vector space. Closure under addition and scalar multiplication follows from the linearity of the equations and the fact that the right-hand side is zero. This example connects vector spaces directly to systems of equations and linear algebraic modeling.
4 Subspaces
A subspace is a smaller vector space contained within a larger one, using the same addition and scalar multiplication. Subspaces are the natural setting for studying restrictions, solution sets, and linear spans.
4.1 Definition of a subspace
A subset of a vector space is a subspace if it is itself a vector space under the inherited operations. In practice, this means it must contain the zero vector and be closed under addition and scalar multiplication. Subspaces preserve linear structure.
4.2 Subspace test
A common criterion for identifying a subspace is the subspace test. A nonempty subset is a subspace if every linear combination of two of its vectors remains in the subset, or equivalently if it is closed under addition and scalar multiplication. This test is often the quickest way to verify the subspace property.
4.3 Intersection and sum of subspaces
The intersection of subspaces is always a subspace, since any vector common to both inherits closure from each. The sum of two subspaces consists of all vectors that can be written as a sum of one vector from each subspace. These constructions help describe how subspaces overlap and combine.
4.4 Span of a set
The span of a set is the collection of all vectors that can be formed from finite linear combinations of the set’s elements. It is the smallest subspace containing the set. Span provides a way to generate subspaces from a chosen collection of vectors.
4.4.1 Linear combinations
A linear combination is an expression obtained by multiplying vectors by scalars and adding the results. Such expressions encode the basic operations of linear algebra. Many central questions ask which vectors can be represented as linear combinations of others.
4.4.2 Spanning sets
A spanning set is a collection of vectors whose span is the entire space, or a specified subspace. Spanning sets need not be minimal; they may contain redundancy. They are useful for constructing spaces and for representing vectors in terms of chosen generators.
5 Linear independence
Linear independence describes whether vectors contain redundancy. It is a foundational idea for determining whether a set can serve as a coordinate system or basis.
5.1 Linearly independent sets
A set of vectors is linearly independent if no nontrivial linear combination of them equals the zero vector. In such a set, each vector contributes genuinely new information. Independence is a key criterion for efficient representation.
5.2 Linearly dependent sets
A set is linearly dependent if at least one vector can be expressed as a linear combination of the others. Dependence indicates redundancy within the set. Dependent collections are common, especially when too many vectors are chosen for a space of limited dimension.
5.3 Dependence relations
A dependence relation is a nontrivial linear equation among vectors that sums to zero. Such a relation can often be rearranged to express one vector in terms of the others. Dependence relations are used to identify unnecessary vectors and simplify spanning sets.
5.4 Criteria for independence
Independence can be checked in several ways, depending on the setting. For coordinate vectors, systems of equations and determinants may be useful; for more general spaces, direct analysis of linear combinations is often required. The appropriate criterion depends on the form of the vectors involved.
6 Bases and dimension
Bases and dimension provide a compact description of the size and structure of a vector space. They are central to understanding how vectors can be represented uniquely.
6.1 Basis of a vector space
A basis is a linearly independent spanning set. Every vector in the space can be written as a linear combination of basis vectors, and this representation is unique. Bases serve as coordinate frameworks for vector spaces.
6.2 Coordinate representation
Once a basis is chosen, each vector is identified by the list of scalars used in its basis expansion. These lists are called coordinates relative to the basis. Coordinate representation allows abstract vectors to be manipulated as tuples of scalars.
6.3 Dimension
The dimension of a vector space is the number of vectors in any basis, when that number is finite. All bases of a given vector space have the same cardinality, so dimension is well defined. It measures the number of independent directions or degrees of freedom in the space.
6.4 Finite-dimensional and infinite-dimensional spaces
A vector space is finite-dimensional if it has a basis with finitely many elements. Otherwise, it is infinite-dimensional. Infinite-dimensional spaces occur naturally in analysis and function theory, where no finite list of vectors can generate the whole space.
6.5 Basis extension and replacement
A linearly independent set can often be extended to a basis, and a spanning set can often be reduced to a basis. These processes reflect the flexibility of vector space structure. They are especially important in proofs involving dimension and the construction of coordinates.
7 Linear transformations
Linear transformations are functions between vector spaces that preserve vector addition and scalar multiplication. They capture the notion of structure-preserving maps and are among the most important objects in linear algebra.
7.1 Definition of a linear map
A linear map sends sums to sums and scalar multiples to scalar multiples. This means that the image of a linear combination is the corresponding linear combination of images. Linearity ensures that the transformation respects the vector space structure.
7.2 Kernel and image
The kernel of a linear map is the set of vectors mapped to the zero vector. The image is the set of all outputs attained by the map. Both are subspaces, and they describe the behavior of the map in terms of lost information and reachable values.
7.3 Rank and nullity
The rank of a linear map is the dimension of its image, while the nullity is the dimension of its kernel. These numerical invariants summarize important features of the map. They are connected by the rank-nullity relationship in finite-dimensional settings.
7.4 Matrix representation
A linear map between finite-dimensional vector spaces can be represented by a matrix once bases are chosen. The matrix encodes the action of the map on basis vectors and then on all vectors by linearity. This translation between abstract maps and concrete arrays is a core technique in linear algebra.
7.5 Change of basis
Changing the basis changes the coordinate description of vectors and linear maps without changing the underlying objects. The same vector may have different coordinate lists in different bases. Change of basis formulas allow one to compare these representations systematically.
8 Direct sums and decompositions
Direct sums describe ways of building a vector space from smaller subspaces that fit together without overlap. They are useful for breaking problems into simpler parts.
8.1 Internal direct sum
An internal direct sum occurs when a space is the sum of subspaces whose intersection is trivial. In that case, every vector in the space can be expressed uniquely as a sum of components from the subspaces. This gives a structured decomposition within a single vector space.
8.2 External direct sum
An external direct sum constructs a new vector space from a collection of vector spaces by combining their elements into tuples. Addition and scalar multiplication are defined componentwise. This construction is useful for assembling larger spaces from known ones.
8.3 Complementary subspaces
Two subspaces are complementary if their sum is the whole space and their intersection is only the zero vector. Complementary subspaces provide a clean way to split vectors into independent parts. Such decompositions often simplify analysis and computation.
8.4 Decomposition of vectors
When a space is decomposed into direct summands, each vector can be written uniquely as a sum of components from the pieces. This decomposition often clarifies the geometry or algebra of the space. It is especially valuable in solving equations and studying projections.
9 Quotient spaces
Quotient spaces formalize the idea of identifying vectors that differ by an element of a subspace. They are widely used in algebra and linear analysis.
9.1 Cosets of a subspace
Given a subspace, a coset is formed by adding a fixed vector to every element of the subspace. All vectors in the same coset differ by an element of the subspace. Cosets partition the original space into equivalence classes.
9.2 Definition of the quotient space
The quotient space is the set of all cosets of a subspace, with operations induced from the original vector space. It becomes a vector space in its own right. The quotient measures the original space modulo the chosen subspace.
9.3 Canonical projection
The canonical projection sends each vector to its coset in the quotient space. It is a linear map that collapses the subspace to zero. This projection is the natural bridge between a vector space and its quotient.
9.4 First isomorphism theorem
The first isomorphism theorem states that the image of a linear map is naturally isomorphic to the quotient of the domain by the kernel. This result explains how linear maps factor through quotient spaces. It is one of the central structural theorems of linear algebra.
10 Dual spaces
The dual space consists of all linear functionals on a vector space. It packages the ways in which vectors can be measured linearly.
10.1 Linear functionals
A linear functional is a linear map from a vector space to its field of scalars. Such maps assign a scalar value to each vector while preserving linear combinations. They are important in analysis, geometry, and optimization.
10.2 Dual basis
If a vector space has a basis, the dual basis is the corresponding family of linear functionals that pick out the coordinates of vectors relative to that basis. Each functional evaluates to one on a matching basis vector and to zero on the others. Dual bases make coordinate extraction precise.
10.3 Bidual space
The bidual is the dual of the dual space. There is a natural map from a vector space into its bidual, associating each vector with the functional that evaluates dual elements on it. In finite dimensions, this map is an isomorphism.
10.4 Annihilators
The annihilator of a set is the collection of linear functionals that vanish on every vector in that set. Annihilators connect subspaces of a vector space with subspaces of its dual. They are useful in characterizing orthogonality-like conditions in purely algebraic terms.
11 Additional structures and special classes
Vector spaces often carry extra structure that enriches their theory. These additions introduce notions such as length, angle, continuity, and convergence.
11.1 Inner product spaces
An inner product space is a vector space equipped with an inner product, which assigns a scalar to pairs of vectors. This structure allows one to define lengths, angles, and orthogonality. It is the foundation for much of geometry and spectral theory.
11.2 Normed vector spaces
A normed vector space has a norm that assigns a nonnegative length to each vector. Norms induce distance and permit the study of convergence and continuity. They are central in functional analysis.
11.3 Topological vector spaces
A topological vector space combines vector space operations with a topology compatible with addition and scalar multiplication. This compatibility permits the study of continuity in a broad setting. Many spaces of functions and distributions are treated in this way.
11.4 Modules as generalizations
Modules generalize vector spaces by allowing scalars from a ring rather than a field. Because rings may lack multiplicative inverses for nonzero elements, modules can behave differently from vector spaces. They broaden the scope of linear methods in algebra.
12 Applications
Vector spaces appear throughout mathematics and its applications. Their abstract structure makes them a versatile tool for modeling and computation.
12.1 Geometry and affine spaces
In geometry, vector spaces describe directions, displacements, and coordinate systems. They underlie the study of lines, planes, and higher-dimensional shapes. Affine spaces build on vector spaces while separating points from displacement vectors.
12.2 Differential equations
Solutions of linear differential equations often form vector spaces or related affine spaces. Superposition principles depend on linearity, allowing sums of solutions to remain solutions in homogeneous cases. This viewpoint is fundamental in applied mathematics.
12.3 Physics and mechanics
Vector spaces are used to represent states, forces, velocities, and other physical quantities. Linear models simplify the analysis of motion, equilibrium, and wave behavior. In mechanics, the language of vectors and linear maps is especially pervasive.
12.4 Data analysis and computer science
Vector spaces appear in data representation, signal processing, graphics, and machine learning. Lists of features can be treated as coordinate vectors, making linear methods natural for computation and classification. In computer science, vector space ideas also support algorithms for encoding, transformation, and dimensional reduction.