R1×N

R1×n denotes the set of all real 1 by n matrices, i.e., row vectors of length n. An element of R1×n can be written as [x1 x2 ... xn], where each xi is a real number. Equivalently, R1×n is the space of all 1×n real matrices M = (x1, x2, ..., xn).

R1×n is a real vector space under the usual operations of componentwise addition and scalar multiplication. Its dimension is n, and a standard basis is the set of row vectors that have a 1 in one position and 0 in all others: [1 0 ... 0], [0 1 0 ... 0], ..., [0 ... 0 1].

This space is naturally identified with R^n via transposition: the map that sends a row vector a = [a1 ... an] to the column vector a^T ∈ R^n is a linear isomorphism. Under this identification, the usual dot product on R^n corresponds to the bilinear form that takes a ∈ R1×n and b ∈ Rn×1 to the scalar a b ∈ R.

Matrix multiplication extends the utility of R1×n: if a ∈ R1×n and B ∈ Rn×m, then aB ∈ R1×m is a row vector, the ordinary left multiplication of a row by a matrix. If v ∈ Rn×1 is a column vector, then av ∈ R1×1 is the dot product of a with v.

R1×n is commonly used to represent row vectors, coefficients of linear forms, or covectors in certain conventions, and its properties mirror those of R^n under the standard identification.