Vector spaces are playgrounds closed under scale and add
A vector space—or linear space—is a set of vectors you can add and stretch by scalars while staying inside the set. The idea generalizes arrows for force and velocity into pure structure. Dimension counts how many independent directions you need.
Formally, you start with a field F of scalars and a set of vectors that is not empty, plus two operations: adding two vectors to get a third, and multiplying a vector by a scalar from F. Eight axioms govern them. The four about addition amount to saying the vectors form an abelian group; the other four say that scaling behaves like a ring homomorphism from the field into that group's endomorphisms, which lets mathematicians sum it all up as a module over a field. Consequences follow quickly: zero times any vector is the zero vector, and multiplying by minus one gives the negative. When scalars are real numbers the space is called real, when complex, complex.
Pick any vectors and form sums of them weighted by scalars, and you have linear combinations. A set is linearly independent when none of its members can be built from the others, and it spans a space if its combinations reach everything. A basis does both. Every vector space has one, and all its bases have the same size, which is the dimension. Relative to a basis, each vector has a unique list of coordinates, so computations on abstract vectors turn into arithmetic on n-tuples. A subspace is a subset closed under both operations, and any intersection of subspaces is again a subspace.
Infinite dimensions bring surprises. Infinite bases, called Hamel bases, exist only thanks to the axiom of choice, and usually none can be written down: the real numbers form a vector space over the rationals for which nobody knows a specific basis. Richer versions add structure, from polynomial rings and Lie algebras to Hilbert and Banach spaces.
The idea grew out of geometry. Descartes and Fermat founded analytic geometry around 1636, Bolzano defined operations on points and lines in 1804, and Möbius introduced barycentric coordinates in 1827. Hamilton's quaternions, Grassmann's 1844 work on independence and dimension, and Cayley's 1857 matrix notation followed, until Giuseppe Peano gave the modern definition in 1888 under the name linear systems.
Source: Vector space
