A vector space is a mathematical structure whose elements, called vectors, can be added together and multiplied by scalars from a specified field. These operations satisfy rules that generalize familiar calculations with arrows and coordinate lists. Vectors need not be geometric objects: functions, polynomials, and matrices can also form vector spaces. The concept provides the basic setting for linear algebra, allowing the same methods to describe many different kinds of mathematical objects. (math.brown.edu)
Definition and axioms
A vector space over a field (F) consists of a set (V), an addition operation (V\times V\to V), and scalar multiplication (F\times V\to V). Common scalar fields are the real numbers (\mathbb R) and complex numbers (\mathbb C). The operations are closed: adding two vectors or multiplying a vector by a scalar always produces another element of (V). (math.brown.edu)
For all (u,v,w\in V) and (a,b\in F), the following axioms hold:
- Commutativity of addition: (u+v=v+u).
- Associativity of addition: ((u+v)+w=u+(v+w)).
- Additive identity: a zero vector (0_V) satisfies (v+0_V=v).
- Additive inverses: each (v) has an element (-v) satisfying (v+(-v)=0_V).
- Compatibility of scalar multiplication: (a(bv)=(ab)v).
- Scalar identity: (1v=v).
- Distributivity over vector addition: (a(u+v)=au+av).
- Distributivity over scalar addition: ((a+b)v=av+bv).
These rules distinguish vector addition from scalar multiplication; multiplication of two vectors is not required. (math.mit.edu)
Examples and the role of scalars
The coordinate space (F^n) consists of ordered lists of (n) scalars. Addition and scalar multiplication act component by component. For example, in (\mathbb R^2), [ (1,2)+(3,-1)=(4,1),\qquad 2(1,2)=(2,4). ] The space containing only the zero vector is also a vector space. (math.brown.edu)
The set of all (m\times n) matrices over (F) forms a vector space under entrywise addition and scalar multiplication. The polynomials of degree at most (d), including the zero polynomial, form another example. More generally, all functions from a fixed set into (F) form a vector space under pointwise operations. These examples show why “vector” describes an algebraic role rather than a particular shape or notation. (linear.axler.net)
The scalar field is part of the structure, not an incidental choice. The complex numbers constitute a one-dimensional space over (\mathbb C), with basis (1), but a two-dimensional space over (\mathbb R), with basis (1,i). (linear.axler.net)
Linear combinations, bases, and dimension
A linear combination of vectors (v_1,\ldots,v_k) is an expression [ a_1v_1+\cdots+a_kv_k,\qquad a_j\in F. ] The span of a collection is the set of all its finite linear combinations. Vectors are linearly independent when a combination equals zero only if every coefficient is zero. Otherwise, they are linearly dependent. (ocw.mit.edu)
A basis is a linearly independent collection that spans the space. Every vector has a unique expression as a finite linear combination of basis vectors. For a finite ordered basis (b_1,\ldots,b_n), [ v=x_1b_1+\cdots+x_nb_n. ] The coefficients form the coordinate list of (v) relative to that basis. Changing the basis changes the coordinates, not the underlying vector. (linear.axler.net)
The dimension of a finite-dimensional space is the number of vectors in any basis; all its bases have the same size. Thus (F^n) has dimension (n). If no finite collection spans a space, it is infinite-dimensional. The space of all polynomials provides an example: arbitrarily high powers cannot be generated by finitely many polynomials. (ocw.mit.edu)
Subspaces and linear maps
A linear subspace is a subset that is itself a vector space under the inherited operations. It must contain zero and be closed under addition and scalar multiplication. Lines and planes through the origin are familiar examples. A translated plane not containing the origin is instead an affine subset. Solutions of a homogeneous system (Ax=0) form a subspace; if (Ax=b) has a solution (x_0), its solutions are (x_0+\ker A). (ocw.mit.edu)
A linear map (T:V\to W), between spaces over the same field, preserves linear combinations: [ T(au+bv)=aT(u)+bT(v). ] Its kernel consists of vectors mapped to zero, and its image consists of attained outputs. Both are subspaces. For finite-dimensional (V), the rank–nullity theorem states [ \dim V=\dim\ker T+\dim\operatorname{im}T. ] After bases are chosen, linear maps between finite-dimensional spaces are represented by matrices. A bijective linear map is an isomorphism; every (n)-dimensional space over (F) is isomorphic to (F^n). (linear.axler.net)
Additional structures and constructions
The vector-space axioms alone define neither lengths nor angles. An inner product supplies additional structure for orthogonality and induces a norm. A normed vector space has a specified notion of length, while a Hilbert space is an inner-product space complete in its induced norm. A compatible topology allows convergence and continuity to be studied alongside the algebraic operations. (math.brown.edu)
The dual space (V^*) consists of linear maps from (V) to its scalar field. A quotient space (V/U) identifies vectors whose difference belongs to a subspace (U). The tensor product provides a vector-space framework for multilinear maps, extending linearity to several arguments. These constructions retain the vector-space axioms while organizing functionals, equivalence classes, and multilinear relationships. (math.mit.edu)