Rm×1

Rm×1, written as R^{m×1}, denotes the set of real m-by-1 column vectors. An element x ∈ R^{m×1} is a column vector with m real components, often written as x = (x1, x2, ..., xm)^T. This space is a real vector space of dimension m and is isomorphic to R^m. The standard basis consists of the vectors e1, e2, ..., em, where ei has a 1 in the i-th position and 0 elsewhere.

Notation can vary; R^{m×1} emphasizes the column structure, while R^m is commonly used for the same underlying set. By contrast, row vectors are elements of R^{1×m}. The column form is the natural orientation for matrix-vector multiplication, where A ∈ R^{k×m} maps x ∈ R^{m×1} to Ax ∈ R^{k×1}.

Operations in R^{m×1} are defined componentwise: for x, y ∈ R^{m×1} and scalar α ∈ R, x + y and αx are computed by adding or scaling each component. The inner product is defined as x^T y = ∑i xi yi, and the Euclidean norm is ||x|| = sqrt(x^T x) = sqrt(∑i xi^2). These give the familiar Euclidean geometry on R^{m}.

Applications include representing feature vectors, data points, and parameter vectors in linear systems, optimization, statistics, and machine learning. Vectors in R^{m×1} serve as inputs to linear maps, with properties governed by the associated matrices. The space is foundational in many mathematical and applied contexts, and its structure underpins the standard coordinate-based approach to multidimensional problems.