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

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, ..., vn), 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
 * $$c(v_1,\ldots, v_n) = (c v_1, \ldots, c v_n)\text{.}$$

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

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