A linear algebra over the field of real numbers R is, by definition, a linear space A over R together with a bilinear map A2 → A, the algebra product.

Examples include R itself, the field of complex numbers, C, consisting of the linear space R2 with the product (a, b)(c, d) = (ac − bd, ad + bc), the double field2R consisting of the linear space R2 with the product (a, b)(c, d) = (ac, bd), and the full matrix algebraR(n) of all n × n matrices with real entries, with matrix multiplication as the product.

An algebra A may, or may not, have a unit element, and the product need be neither commutative nor associative, though it is usual to mention explicitly any failure of associativity. The unit element, if it exists, will normally be denoted by 1(A) or simply, where no confusion need arise, by 1, the map R → A, λ ↦ λ1(A) being injective. (The notation 1A is reserved for the identity map on A.)

All the above examples are associative and have a unit element, and all are commutative, with the exception of the matrix algebra R(n), with n > 1. The double field 2R is often identified with the subalgebra of R(2) consisting of the diagonal 2 × 2 matrices, the unit element being denoted by 21.