n×1

An N×1, in linear algebra, denotes a matrix with N rows and 1 column. Equivalently, it is an N-dimensional column vector, often written as x = [x1; x2; ...; xN], with x1 through xN being its components. Such a matrix is an element of the real or complex space R^N (or C^N, respectively) depending on the underlying field.

As a matrix, an N×1 has several basic properties. Its transpose is a 1×N row vector. Addition and scalar multiplication are defined componentwise, so two N×1 matrices can be added only if they have the same dimension, and multiplying by a scalar scales each component. The rank of an N×1 matrix is 0 if the vector is the zero vector, and 1 otherwise; the column space is the set of all multiples of the vector, i.e., the span of that single vector.

Operations involving N×1 vectors are common. The dot product of two N×1 vectors x and y is given by x^T y, a scalar. The outer product x y^T of two N×1 vectors yields an N×N matrix. If A is N×1 and B is 1×M, their product A B is an N×M matrix, representing an outer product expanded across the chosen dimensions.

Applications of N×1 vectors include representing coordinates in Euclidean space, feature vectors in data analysis, and the images of scalars under linear transformations. In computations, an N×1 vector is the typical form for data samples and state representations, with N indicating dimensionality. When N = 1, the N×1 matrix reduces to a 1×1 scalar.