Vector space

A set equipped with vector addition and scalar multiplication over a field, satisfying axioms that make linear combinations and linear structure well-defined.

Field (mathematics): A set with addition and multiplication in which nonzero elements have multiplicative inverses, satisfying field axioms. Scalars come from a field, whose operations make scalar multiplication coherent.
Span (linear algebra): The set of all linear combinations of a given set of vectors. Spanning measures which vectors can be built from a chosen collection.
System of linear equations: A collection of equations in which each equation is linear in the unknowns. Solutions form an affine subset, or a vector space when the system is homogeneous.
Module (mathematics): An abelian group with scalar multiplication by elements of a ring, satisfying module axioms. Modules generalize vector spaces by allowing scalars from rings that may lack inverses.
Group (mathematics): A set with an associative operation, an identity element, and inverses for every element. Vector addition makes each vector space an abelian group.
Linear independence: A property of vectors for which only the trivial linear combination equals zero. Independence rules out redundant vectors in a vector-space description.
Matrix (mathematics): A rectangular array of entries representing data or a linear transformation. Matrices encode linear maps after bases are chosen.
Affine space: A space of points whose differences are vectors, without a distinguished zero point. An affine space has vector-like displacements but no canonical zero vector.
Abelian group: A group whose operation is commutative. The vector-addition axioms require vectors to add commutatively.
Dimension (vector space): The number of vectors in a basis of a finite-dimensional vector space. Dimension counts the independent directions available in a finite-dimensional space.

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