1×nmatrix

1×nmatrix refers to a one-by-n matrix, commonly known as a row vector. It consists of a single row with n entries, typically written as A = [a1 a2 ... an], and it represents a linear object in R^{1×n} when the entries are real. The transpose A^T is an n×1 column vector, linking row and column representations.

Operations and properties: If B is an n×m matrix, the product AB yields a 1×m row vector. For any x in R^n, the product A x is a scalar equal to the dot product of A and x, namely ∑_{i=1}^n ai xi. A can be scaled by any scalar α, giving αA. The rank of a 1×n matrix is at most 1, and its row space is one-dimensional. As a functional, A defines a linear map from R^n to R by x ↦ A x.

Representation in software: In numerical computing, a 1×n matrix is often used to represent a row vector of features or observations. Some libraries and languages treat these as a distinct matrix shape (1 by n) rather than a simple one-dimensional array, facilitating matrix-oriented operations such as multiplication with n×m matrices.

Applications: Row vectors are frequently used to store data instances, coefficient vectors in linear models, or to perform dot-product computations in machine learning and statistics. They are also convenient for expressing linear functionals and for presenting results as a single-row transformation of input data.

Terminology: The term can be described as a 1×n matrix, a row vector, or a 1-by-n matrix; in software libraries it may be implemented as a specific matrix shape rather than a flat array.