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Overview.
The word vector has several distinct, but interrelated meanings. The present entry is an overview and discussion of these concepts, with links at the end to more detailed definitions.

A list vector (follow the link to the formal definition) is a finite list of numbers^{1}^{1}Infinite vectors arise in areas such as functional analysis and quantum mechanics, but require a much more complicated and sophisticated theory.. Most commonly, the vector is composed of real numbers, in which case a list vector is just an element of $\mathbb{R}^{n}$. Complex numbers are also quite common, and then we speak of a complex vector, an element of $\mathbb{C}^{n}$. Lists of ones and zeroes are also utilized, and are referred to as binary vectors. More generally, one can use any field $\mathbb{K}$, in which case a list vector is just an element of $\mathbb{K}^{n}$.

A physical vector (follow the link to a formal definition and indepth discussion) is a geometric quantity that correspond to a linear displacement. Indeed, it is customary to depict a physical vector as an arrow. By choosing a system of coordinates a physical vector $\mathbf{v}$, can be represented by a list vector $(v^{1},\ldots,v^{n})^{{\scriptscriptstyle\mathrm{T}}}$. Physically, no single system of measurement cannot be preferred to any other, and therefore such a representation is not canonical. A linear change of coordinates induces a corresponding linear transformation of the representing list vector.
In most physical applications vectors have a magnitude as well as a direction, and then we speak of a Euclidean vector. When lengths and angles can be measured, it is most convenient to utilize an orthogonal system of coordinates. In this case, the magnitude of a Euclidean vector $\mathbf{v}$ is given by the usual Euclidean norm of the corresponding list vector,
$\\mathbf{v}\=\sqrt{{\textstyle\sum_{i}}(v^{i})^{2}}\;.$ This definition is independent of the choice of orthogonal coordinates.

An abstract vector is an element of a vector space. An abstract Euclidean vector is an element of an inner product space. The connection between list vectors and the more general abstract vectors is fully described in the entry on frames.
Essentially, given a finite dimensional abstract vector space, a choice of a coordinate frame (which is really the same thing as a basis) sets up a linear bijection between the abstract vectors and list vectors, and makes it possible to represent the one in terms of the other. The representation is not canonical, but depends on the choice of frame. A change of frame changes the representing list vectors by a matrix multiplication.
We also note that the axioms of a vector space make no mention of lengths and angles. The vector space formalism can be enriched to include these notions. The result is the axiom system for inner products.
Why do we bother with the “barebones” formalism of lengthless vectors? The reason is that some applications involve velocitylike quantities, but lack a meaningful notion of speed. As an example, consider a multiparticle system. The state of the system is represented as a point in some manifold, and the evolution of the system is represented by velocity vectors that live in that manifold’s tangent space. We can superimpose and scale these velocities, but it is meaningless to speak of a speed of the evolution.
Discussion.
What is a vector? This simple question is surprisingly difficult to answer. Vectors are an essential scientific concept, indispensable for both the physicist and the mathematicians. It is strange then, that despite the obvious importance, there is no clear, universally accepted definition of this term.
The difficulty is one of semantics. The term vector is ambiguous, but its various meanings are interrelated. The different usages of vector call for different formal definitions, which are similarly interrelated. List vectors are the most elementary and familiar kind of vectors. They are easy to define, and are mathematically precise. However, saying that a vector is just a list of numbers leads to conceptual difficulties.
A physicist needs to be able to say that velocities, forces, fluxes are vectors. A geometer, and for that matter a pilot, will think of a vector as a kind of spatial displacement. Everyone would agree that a choice of a vector involves multiple degrees of freedom, and that vectors can linearly superimposed. This description of “vector” evokes useful and intuitive understanding, but is difficult to formalize.
The synthesis of these conflicting viewpoints is the modern mathematical notion of a vector space. The key innovation of modern, formal mathematics is the pursuit of generality by means of abstraction. To that end, we do not give an answer to “What is a vector?”, but rather give a list of properties enjoyed by all objects that one may reasonably term a “vector”. These properties are just the axioms of an abstract vector space, or as Forrest Gump[1] might have put it, “A vector is as a vector does.”
The axiomatic approach afforded by vector space theory gives us maximum flexibility. We can carry out an analysis of various physical vector spaces by employing propositions based on vector space axioms, or we can choose a basis and perform the same analysis using list vectors. This flexibility is obtained by means of abstraction. We are not obliged to say what a vector is; all we have to do is say that these abstract vectors enjoy certain properties, and make the appropriate deductions. This is similar to the idea of an abstract class in objectoriented programming.
Surprisingly, the idea that a vector is an element of an abstract vector space has not made great inroads in the physical sciences and engineering. The stumbling block seems to be a poor understanding of formal, deductive mathematics and the unstated, but implicit attitude that
formal manipulation of a physical quantity requires that it be represented by one or more numbers.
Great historical irony is at work here. The classical, Greek approach to geometry was purely synthetic, based on idealized notions like point and line, and on various axioms. Analytic geometry, a la Descartes, arose much later, but became the dominant mode of thought in scientific applications and largely overshadowed the synthetic method. The pendulum began to swing back at the end of the nineteenth century as mathematics became more formal and important new axiomatic systems, such as vector spaces, fields, and topology, were developed. The cost of increased abstraction in modern mathematics was more than justified by the improvement in clarity and organization of mathematical knowledge.
Alas, to a large extent physical science and engineering continue to dwell in the $19^{{\text{th}}}$ century. The axioms and the formal theory of vector spaces allow one to manipulate formal geometric entities, such as physical vectors, without first turning everything into numbers. The increased level of abstraction, however, poses a formidable obstacle toward the acceptance of this approach. Indeed, mainstream physicists and engineers do not seem in any great hurry to accept the definition of vector as something that dwells in a vector space. Until this attitude changes, vector will retain the ambiguous meaning of being both a list numbers, and a physical quantity that transforms with respect to matrix multiplication.
References
 1 R. Zemeckis, “Forrest Gump”, Paramount Pictures.
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