vector

Overview.

The word vector has several distinct, but interrelatedmeanings. The present entry is an overview and discussion of theseconcepts, 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. Most commonly,the vector is composed of real numbers, in which case a list vectoris just an element of ${ℝ}^n$. Complex numbers are also quitecommon, and then we speak of a complex vector, an element of${ℂ}^n$. Lists of ones and zeroes are also utilized, and arereferred to as binary vectors. More generally, one can use anyfield $𝕂$, in which case a list vector is just an element of${𝕂}^n$.

• A physical vector (follow the link to a formal definitionand in-depth discussion) is a geometric quantity that correspond toa linear displacement. Indeed, it is customary to depict a physicalvector as an arrow. By choosing a system of coordinates a physicalvector $𝐯$, can be represented by a list vector${(v^1,\mathrm{\dots },v^n)}^{\mathrm{T}}$. Physically, no single system ofmeasurement cannot be preferred to any other, and therefore such arepresentation is not canonical. A linear change of coordinatesinduces a corresponding linear transformation of the representinglist vector.

  In most physical applications vectors have a magnitude as well as adirection, and then we speak of a Euclidean vector. When lengthsand angles can be measured, it is most convenient to utilize anorthogonal system of coordinates. In this case, the magnitude of aEuclidean vector $𝐯$ is given by the usual Euclidean norm of thecorresponding list vector,

  $$\parallel 𝐯\parallel =\sqrt{{\sum }_i{(v^i)}^2}.$$

  Thisdefinition is independent of the choice of orthogonal coordinates.

• An abstract vector is an element of a vector space. Anabstract Euclidean vector is an element of an inner product space.The connectionbetween list vectors and the more general abstract vectors is fullydescribed in the entry on frames (http://planetmath.org/Frame).

  Essentially, given a finite dimensional abstract vector space, achoice of a coordinate frame (which is really the same thing as abasis) sets up a linear bijection between the abstract vectors andlist vectors, and makes it possible to represent the one in terms ofthe other. The representation is not canonical, but depends on thechoice of frame. A change of frame changes the representing listvectors by a matrix multiplication.

  We also note that the axioms of a vector space make no mention oflengths and angles. The vector space formalism can be enriched toinclude these notions. The result is the axiom system for innerproducts.

  Why do we bother with the “bare-bones” formalism of length-lessvectors? The reason is that some applications involve velocity-likequantities, but lack a meaningful notion of speed. As an example,consider a multi-particle system. The state of the system isrepresented as a point in some manifold, and the evolution of thesystem is represented by velocity vectors that live in thatmanifold’s tangent space. We can superimpose and scale thesevelocities, but it is meaningless to speak of a speed of theevolution.

Discussion.

What is a vector? This simple question is surprisingly difficult toanswer. Vectors are an essential scientific concept, indispensablefor both the physicist and the mathematicians. It is strange then,that despite the obvious importance, there is no clear, universallyaccepted definition of this term.

The difficulty is one of semantics. The term vector isambiguous, but its various meanings are interrelated. The differentusages of vector call for different formal definitions, whichare similarly interrelated. List vectors are the most elementary andfamiliar kind of vectors. They are easy to define, and aremathematically precise. However, saying that a vector is just a listof numbers leads to conceptual difficulties.

A physicist needs to be able to say that velocities, forces, fluxesare vectors. A geometer, and for that matter a pilot, will think of avector as a kind of spatial displacement. Everyone would agree that achoice of a vector involves multiple degrees of freedom, and thatvectors can linearly superimposed. This description of “vector”evokes useful and intuitive understanding, but is difficult toformalize.

The synthesis of these conflicting viewpoints is the modernmathematical notion of a vector space. The key innovation of modern,formal mathematics is the pursuit of generality by means ofabstraction. To that end, we do not give an answer to “What is avector?”, but rather give a list of properties enjoyed by all objectsthat one may reasonably term a “vector”. These properties are justthe 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 usmaximum flexibility. We can carry out an analysis of various physicalvector spaces by employing propositions based on vector space axioms,or we can choose a basis and perform the same analysis using listvectors. This flexibility is obtained by means of abstraction. Weare not obliged to say what a vector is; all we have to do issay that these abstract vectors enjoy certain properties, and make theappropriate deductions. This is similar to the idea of an abstractclass in object-oriented programming.

Surprisingly, the idea that a vector is an element of an abstractvector space has not made great inroads in the physical sciences andengineering. The stumbling block seems to be a poor understanding offormal, deductive mathematics and the unstated, but implicit attitudethat

formal manipulation of a physical quantity requires that it berepresented by one or more numbers.

Great historical irony is at work here. The classical, Greek approachto geometry was purely synthetic, based on idealized notions likepoint and line, and on various axioms. Analytic geometry, a lahttp://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Descartes.htmlDescartes,arose much later, but became the dominant mode of thought inscientific applications and largely overshadowed the synthetic method.The pendulum began to swing back at the end of the nineteenth centuryas mathematics became more formal and important new axiomatic systems,such as vector spaces, fields, and topology, were developed. The costof increased abstraction in modern mathematics was more than justifiedby the improvement in clarity and organization of mathematicalknowledge.

Alas, to a large extent physical science and engineering continue todwell in the ${19}^{\text{th}}$ century. The axioms and the formal theory ofvector spaces allow one to manipulate formal geometric entities, suchas physical vectors, without first turning everything into numbers.The increased level of abstraction, however, poses a formidableobstacle toward the acceptance of this approach. Indeed, mainstreamphysicists and engineers do not seem in any great hurry to accept thedefinition of vector as something that dwells in a vector space.Until this attitude changes, vector will retain the ambiguousmeaning of being both a list numbers, and a physical quantity thattransforms with respect to matrix multiplication.