Paul D. Anderson

I am a software engineer living and working in the Seattle area.

Any opinions expressed are entirely my own.

A vector space, also called a linear space, is a mathematical structure consisting of a set of elements, called vectors, and two binary operations: vector addition and scalar multiplication. These operations have to satisfy certain axioms, listed below.

An example of a vector space is three-dimensional Euclidean space, R³. In this space, a vector consists of an ordered triplet of real numbers, (x, y, z). The components of the vector, x, y, and z, specify the three-dimensional coordinates of the vector, and can represent, for example, position. Vector addition and scalar multiplication for this space are defined component-wise:

$(x_{1},y_{1},z_{1})+(x_{2},y_{2},z_{2})=(x_{1}+x_{2},y_{1}+y_{2},z_{1}+z_{2});\quad a\times (x,y,z)=(a\times x,a\times y,a\times z).$

Vector spaces are characterized, in part, by their dimension and by the nature of the scalars used. Generally, scalars must be elements of a mathematical structure called a field. Common examples of fields are the real and complex numbers. Many useful vector spaces (but by no means all) are defined over the real and complex numbers. The dimension of a space is, roughly, the number of independent directions in the space. A vector space can have a single dimension, or the number of dimensions can be very large, even infinite. R³ is a three-dimensional real vector space. (Alternatively, it is a three-dimensional space, or 3-space, over the reals.)

Vector spaces are the subject of linear algebra. The linearity of the space derives from the axioms, the study of sets of linear equations. The axioms of the binary operations preserve linearity and are useful in sol

The theory is further enhanced by introducing on a vector space some additional structure, such as a norm or inner product. Such spaces arise naturally in mathematical analysis, mainly in the guise of infinite-dimensional function spaces whose vectors are functions. Analytical problems call for the ability to decide if a sequence of vectors converges to a given vector. This is accomplished by considering vector spaces with additional data, mostly spaces endowed with a suitable topology, thus allowing the consideration of proximity and continuity issues. These topological vector spaces, in particular Banach spaces and Hilbert spaces, have a richer theory.