User:Jim.belk/Draft:Vector (mathematics)

A field of tangent vectors to the sphere.

In mathematics, especially linear algebra and vector calculus, a vector is any finite list of real numbers:

${\textbf {v}}=(v_{1},v_{2},\ldots ,v_{n})$

The individual numbers $v_{i}\,\!$ are called the components (or coordinates) of the vector. Geometrically, a vector can be interpreted either as a point in n-dimensional Euclidean space, or as a spatial vector with magnitude and direction.

More generally, a vector may be any element of an abstract vector space. This includes vectors whose components are elements of an arbitrary field (such as the complex numbers), and tangent vectors to a manifold, which are the elements of a tangent space. In addition, the word "vector" is sometimes used loosely to refer to any ordered n-tuple whose components are elements of the same set.

Notation

Vectors are most commonly written as n-tuples (v₁, v₂, ..., v_n), or as column vectors:

${\textbf {v}}={\begin{bmatrix}v_{1}\\v_{2}\\\vdots \\v_{n}\end{bmatrix}}$

The latter notation comes from linear algebra, where column vectors are interpreted as matrices with a single column.

In elementary textbooks, vector variables are usually distinguished by writing them in a bold font (${\textbf {v}}\,\!$ instead of $v\,\!$), or by placing an arrow above the name of the variable (as in ${\vec {v}}$). This distinction becomes less common in higher mathematics, where vectors are more often distinguished by restricting vector variables to a certain set of letters (such as u, v, and w).

Vector operations

The primary vector operations are vector addition

$(v_{1},\ldots ,v_{n})+(w_{1},\ldots ,w_{n})=(v_{1}+w_{1},\ldots ,v_{n}+w_{n})$

and scalar multiplication

$c(v_{1},\ldots ,v_{n})=(cv_{1},\ldots ,cv_{n}){\text{.}}$

Using these operations, the set of vectors with n components satisfies all the axioms for a

More stuff

The set of all vectors with n components is denoted Rⁿ (often written $\mathbb {R} ^{n}$), and is a model for n-dimensional Euclidean space. Geometrically, vectors can be interpreted either as points in n-dimensional space, or as spatial vectors with magnitude and direction.

More generally, a vector may refer to any element of a vector space. These include:

• complex vectors, whose components are complex numbers,
• vectors with components from any field,
• tangent vectors to a manifold, which are elements of a tangent space, and
• vectors from infinite-dimensional vector spaces, such as a function space.

The word "vector" is also sometimes used loosely as a synonym for ordered n-tuple.

Notation

In elementary texts, vectors are often denoted with bold, or