1×M

1×M refers to a matrix with one row and M columns, equivalently a row vector of length M. It is defined over a field (such as the real or complex numbers) and its entries are typically denoted a1, a2, ..., aM.

Algebraic properties and operations: A 1×M matrix A can be multiplied on the right by an M×N matrix B to produce a 1×N matrix AB. In particular, multiplying A by an M×1 column vector x yields a 1×1 scalar equal to the dot product of the row A and x, namely a1x1 + a2x2 + ... + aMxM. The transpose A^T is an M×1 column vector. The row can also be multiplied on the left by a compatible matrix, for example by a k×1 matrix on the left yields a k×M matrix.

Structure and interpretation: The set of all 1×M matrices over a field forms a vector space of dimension M, naturally isomorphic to the M-dimensional coordinate space F^M. The rank of a nonzero 1×M matrix is 1, since its row is nonzero.

Examples and applications: For M = 3, the row A = [a b c] is a 1×3 matrix. A times a 3×1 column vector [d e f]^T equals ad + be + cf. In linear algebra, 1×M row vectors commonly represent coefficients of linear functionals or serve as feature rows in data matrices. They contrast with M×1 column vectors, which represent coordinate columns; a row vector and a column vector can be combined via the inner product or outer product as needed.

Extensions: The concept generalizes to matrices over any field or ring, and to different multiplications as part of larger matrix algebra.