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

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A field of tangent vectors to the sphere.
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.

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[edit] 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.[1]

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).

[edit] Geometric interpretations

[edit] 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


[edit] More stuff

The set of all vectors with n components is denoted Rn (often written [2] \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.

[edit] Notation

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

[edit] Notes

  1. ^ In linear algebra, the transpose of a column vector is a matrix with only one row, and is called a row vector. To distinguish them from ordered n-tuples, row vectors are usually written without commas: [v1  v2  ยทยทยท  vn].
  2. ^ This notation uses blackboard bold, and is arguably more standard than the notation Rn. The latter is more common on Wikipedia because of technical limitations.