Nby1

Nby1 refers to a matrix with N rows and one column, commonly called a column vector of length N. In linear algebra this object is written as an N×1 matrix and represents a single column of data. Its entries are typically denoted a1, a2, ..., aN, and the vector is often written in column form as [a1, a2, ..., aN]^T, where T denotes transpose.

As column vectors, N×1 objects are fundamental in a variety of operations. If v and w are N×1 vectors, their dot product is obtained by v^T w, yielding a scalar. Matrix multiplication follows standard rules: a matrix A with size N×N times a vector v with size N×1 produces a new vector Av of size N×1. The Euclidean norm of a vector is ||v||2 = sqrt(v^T v). The transpose of a column vector v is a 1×N row vector, v^T.

Representations and usage vary by context. In data science and engineering, an N×1 vector often encodes a feature or measurement set for a single item, with each entry corresponding to a different feature or observation. In solving linear systems, Ax = b, the right-hand side b is frequently an N×1 vector. In programming environments, such as MATLAB or NumPy, column vectors are typically stored with shape (N, 1); 1-dimensional arrays with shape (N,) may be used as a vector but lack an explicit column orientation.

Nby1 contrasts with 1×N row vectors; the distinction matters for matrix multiplication and interpretation. In practice, the concept is ubiquitous wherever a single-column collection of N values is needed.