Nx1

Nx1 denotes a matrix with N rows and 1 column, i.e., a column vector in linear algebra. Here N is a positive integer, and the resulting object has N elements.

A typical Nx1 vector is written as x = [x1; x2; ...; xN], or equivalently as x = (x1, x2, ..., xN)^T, emphasizing its column orientation. In many contexts, x is considered an element of R^N, representing an N-dimensional quantity.

Key properties and operations: addition and subtraction are defined elementwise for vectors of the same size, and scalar multiplication scales every entry. Matrix-vector multiplication is a common use: if A is an MxN matrix and x is an Nx1 vector, then Ax is an Mx1 vector. The transpose of an Nx1 vector is a 1xN row vector. Dot products between two N-dimensional vectors are typically expressed as x^T y, yielding a scalar.

Contexts and usage: Nx1 vectors arise as state or feature vectors in engineering, physics, and machine learning, and as parameter vectors in systems of linear equations or optimization problems. In programming and numerical libraries, shapes are often described as (N, 1) to denote an N-by-1 column vector, distinguishing from a (1, N) row vector.

Relation to 1xN: a 1xN vector is the transpose of an Nx1 vector; they are related but represent different shapes. Special cases include N = 1, where Nx1 is a single scalar placed in a column vector form.