linearspace

Linear space, often called a vector space, is a set V equipped with two operations: addition and scalar multiplication, where the elements of V can be added together and scaled by elements of a field F. These operations satisfy the vector space axioms: closure under addition and scalar multiplication; addition is commutative and associative with a zero vector and additive inverses; scalar multiplication distributes over vector addition and field addition, and is compatible with field multiplication, with 1 acting as the identity on V. When these conditions hold, V is a linear space over F.

Subspaces are subsets that are themselves linear spaces under the same operations; equivalently, W is a subspace of V if it is nonempty and closed under addition and scalar multiplication. A basis is a set of vectors that is linearly independent and spans V or a subspace; the number of vectors in a basis is the dimension. Finite-dimensional spaces have finite bases; infinite-dimensional spaces have infinite bases.

Linear maps (or operators) are functions T: V -> W that preserve addition and scalar multiplication. The kernel consists of vectors mapped to zero; the image consists of vectors obtained as T(v). The rank-nullity theorem states that dim V = dim(ker T) + dim(im T) for finite-dimensional V. If two vector spaces have the same dimension over the same field, they are isomorphic.

Common examples include R^n with ordinary addition and multiplication, polynomials with real coefficients, and function spaces. The structure supports constructions such as direct sum, tensor products, and dual spaces, and underpins much of linear algebra and its applications.