A scalar is a single real or complex number, the simplest object in linear algebra. Building from it, a vector is an ordered list of scalars, which can be visualized geometrically as a directed magnitude in space. A matrix generalizes this further as a rectangular array of numbers arranged in rows and columns, while a tensor is the most general object: a multi-dimensional array whose order ranges from scalars (0-D) to vectors (1-D), matrices (2-D), and higher.
A vector space is a set V equipped with two operations, vector addition and scalar multiplication, satisfying eight axioms: closure under both operations, commutativity and associativity of addition, existence of an additive identity (the zero vector) and additive inverses, a multiplicative identity, and distributivity of scalar multiplication over both vector addition and scalar addition. The difference u − v is then defined as u + (−v), ensuring that subtraction, like addition and scaling, stays inside the space.
Important structural concepts arise from how vectors can be combined. The span of a set of vectors is the set of all their linear combinations \(c_1 v_1 + c_2 v_2 + \cdots + c_n v_n\), and it always forms a subspace. A subspace more generally is any non-empty subset closed under addition and scalar multiplication that contains the zero vector. A set of vectors is linearly independent when no vector in it can be written as a linear combination of the others — equivalently, the only combination that gives the zero vector is the trivial one with all coefficients zero; otherwise the set is linearly dependent.
A basis is a linearly independent spanning set, and the dimension of a vector space is the number of vectors in any basis (every basis has the same size). In \(\mathbb{R}^n\) the standard basis consists of the vectors \(e_1, \ldots, e_n\) with a single 1 in one coordinate and zeros elsewhere. Every vector space has a basis, a statement equivalent to the axiom of choice.