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Vector

Definition

A vector is an element of a vector space. In coordinate spaces such as Rn\mathbb{R}^n, a vector is often represented as an ordered list of coordinates.

A vector written vertically as an n×1n\times1 matrix is called a column vector. A vector written horizontally as a 1×n1\times n matrix is called a row vector.

Interpretation

A vector is an element of a vector space: something that can be added to other vectors of the same space and multiplied by a scalar. In Rn\mathbb{R}^n it is an ordered list of nn numbers, which can be read as a point, an arrow, a displacement or simply a list of data.

The vector v = (3, 2) on a grid drawn as an arrow from the origin, and the same arrow drawn again from another starting point.

The blue arrow from the origin to the point (3,2)(3,2) represents v=(3,2)\boldsymbol{v}=(3,2); the dashed lines show its two coordinates, 33 steps along the x1x_1-axis and 22 along the x2x_2-axis. The lighter arrow starts at (1,12)(1,\tfrac{1}{2}) and makes the same steps, so it represents the same vector: an arrow is fixed by its direction and length, not by where it starts.

Drag the tip of v\boldsymbol{v}: its two coordinates are the steps along the axes, and the same vector can be written as a column or, transposed, as a row.

Writing a vector as a column or as a row matters only in matrix calculations. As an n×1n\times1 matrix, a column vector x\boldsymbol{x} is what a matrix multiplies in a linear system Ax=b\boldsymbol{A}\boldsymbol{x}=\boldsymbol{b}, and row vectors usually appear as transposes x⊤\boldsymbol{x}^{\top} of column vectors.

The general definition is wider than lists of numbers: m×nm\times n matrices and polynomials are vectors too, because they can be added and scaled within their own vector spaces. A common slip is to identify a vector with its list of coordinates. Coordinates depend on a chosen basis; only for the standard basis of Rn\mathbb{R}^n are they the entries of the list itself.

Key Results

  • Vectors can be added and multiplied by elements of the underlying field when they belong to a vector space.
  • The transpose of a column vector is a row vector, and the transpose of a row vector is a column vector.

Examples

  • (123)\begin{pmatrix}1\\2\\3\end{pmatrix} is a column vector in R3\mathbb{R}^3.
  • (123)\begin{pmatrix}1&2&3\end{pmatrix} is a row vector.

Prerequisites

Sources

Keywords: vectors, column vector, row vector, coordinate vector, displacement vector Symbols: real numbers

Last updated: 20 October 2026 03:02