columnvector

A column vector is a matrix with a single column, i.e., an n × 1 object. It represents a point or direction in n-dimensional space and its entries are the coordinates with respect to a chosen basis. An example is v = [3; 4; 5], a 3 × 1 column vector. In many texts, a column vector is denoted by a bold face letter such as v, or by v with a superscript T to indicate transpose, for example v^T is a 1 × n row vector.

Operations involving column vectors include scalar multiplication, componentwise addition, and matrix multiplication. If c is a scalar, then cv scales every entry of v. If u and v are column vectors of the same size, their sum is a column vector whose i-th entry is u_i + v_i. For an m × n matrix A and an n × 1 vector v, the product Av is an m × 1 column vector, representing a linear transformation of v by A.

Related concepts include the row vector, which is the transpose of a column vector; v^T is a 1 × n row vector. The inner product of real column vectors u and v is u^T v, a scalar. The Euclidean norm of v is ||v|| = sqrt(v1^2 + ... + vn^2). The column space of a matrix A is the span of its columns, i.e., the set of all vectors that can be written as Av for some appropriate v.

Column vectors are fundamental in expressing systems of linear equations, linear transformations, and vector coordinates in both real and complex vector spaces.