Vector (math)

Vector

{\overrightarrow {AB}}

{\ displaystyle {\ overrightarrow {AB}}}

\ overrightarrow {AB}

A vector (from Latin vector , “bearing”) is in the simplest case a mathematical object characterized by magnitude and direction. For example, in geometry and in the natural sciences, a vector is a directed segment of a line in Euclidean space (or on a plane) ^[1] .

Examples: radius vector , speed , moment of force . If a coordinate system is specified in space, then the vector is uniquely determined by the set of its coordinates. Therefore, in mathematics, computer science, and other sciences, an ordered set of numbers is also often called a vector. In a more general sense, a vector in mathematics is considered as an element of some vector (linear) space .

It is one of the fundamental concepts of linear algebra . Using the most general definition, vectors turn out to be practically all objects studied in linear algebra, including matrices , tensors , however, if there are these objects in the surrounding context, a vector is understood as a row vector or column vector , tensor of the first rank, respectively. The properties of operations on vectors are studied in vector calculus .

Content

Conventions

Vector represented by a set. $n$ ${\ displaystyle n}$ elements (component) $a_{1},a_{2},\ldots ,a_{n}$ ${\ displaystyle a_ {1}, a_ {2}, \ ldots, a_ {n}}$ designated in the following ways:

\langle a_{1},a_{2},\ldots ,a_{n}\,\rangle ,\ \left(a_{1},a_{2},\ldots ,a_{n}\,\right),\{a_{1},a_{2},\ldots ,a_{n}\,\}

{\ displaystyle \ langle a_ {1}, a_ {2}, \ ldots, a_ {n} \, \ rangle, \ \ left (a_ {1}, a_ {2}, \ ldots, a_ {n} \, \ right), \ {a_ {1}, a_ {2}, \ ldots, a_ {n} \, \}}

.

In order to emphasize that this is a vector (and not a scalar), use the line above, the arrow above, bold or gothic:

{\bar {a}},\ {\vec {a}},\mathbf {a} ,{\mathfrak {A}},\ {\mathfrak {a}}.

{\ displaystyle {\ bar {a}}, \ {\ vec {a}}, \ mathbf {a}, {\ mathfrak {A}}, \ {\ mathfrak {a}}.}

Addition of vectors is almost always indicated by a plus sign:

{\vec {a}}+{\vec {b}}

{\ displaystyle {\ vec {a}} + {\ vec {b}}}

.

Multiplication by a number - simply by writing side by side, without a special sign, for example:

k{\vec {b}}

{\ displaystyle k {\ vec {b}}}

,

and the number is usually written on the left.

Multiplication by a matrix is also indicated by writing next to it, without a special sign, but here the permutation of the factors in the general case affects the result. The action of a linear operator on a vector is also indicated by writing the operator on the left, without a special sign.

History

Intuitively, a vector is understood as an object having a magnitude, direction, and (optionally) an application point. The rudiments of vector calculus appeared along with the geometric model of complex numbers ( Gauss , 1831). Hamilton published the developed operations with vectors as part of his quaternion calculus (the imaginary components of the quaternion formed the vector). Hamilton proposed the term vector itself ( Latin vector , bearing ) and described some operations of vector analysis . This formalism was used by Maxwell in his works on electromagnetism , thereby drawing the attention of scientists to the new calculus. Gibbs (Elements of Vector Analysis) (1880s) soon came out, and then Heaviside (1903) gave vector analysis a modern look. There are no generally accepted vector notations; bold type, a line or arrow above a letter, the Gothic alphabet, etc. are used ^[2] .

In geometry

In geometry, vectors mean directional segments. This interpretation is often used in computer graphics , building lighting maps , using normals to surfaces. Also, using vectors, you can find the areas of various figures, for example, triangles and parallelograms , as well as the volumes of bodies: a tetrahedron and a parallelepiped .
Sometimes a direction is identified with a vector.

A vector in geometry is naturally associated with a transfer ( parallel transfer ), which, obviously, clarifies the origin of its name ( Latin vector , bearing ). Indeed, any directed segment unambiguously defines a parallel translation of a plane or space, and vice versa, parallel translation uniquely defines a single directed segment (unambiguously - if all directed segments of the same direction and length are considered equal - that is, they are considered as free vectors ) .

Interpretation of a vector as a transfer allows a natural and intuitively obvious way to introduce the operation of adding vectors - as a composition (sequential use) of two (or several) transfers; the same applies to the operation of multiplying a vector by a number.

In linear algebra

In linear algebra, a vector is an element of a linear space, which corresponds to the general definition below. Vectors can have a different nature: directed segments, matrices, numbers, functions, and others, however, all linear spaces of the same dimension are isomorphic to each other.
This concept of a vector is most often used in solving systems of linear algebraic equations , as well as when working with linear operators (an example of a linear operator is the rotation operator). Often this definition is expanded by defining a norm or a scalar product (possibly both), after which they operate on normalized and Euclidean spaces, connect the notion of the angle between the vectors with the scalar product, and the notion of the length of the vector with the norm. Many mathematical objects (for example, matrices , tensors , etc.), including those having a structure more general than a finite (and sometimes even countable) ordered list, satisfy the axioms of a vector space , that is, from the point of view of algebra, they are vectors .

In functional analysis

Functional analysis considers functional spaces - infinite - dimensional linear spaces. Their elements may be functions. Based on this representation of a function, the theory of Fourier series is constructed. Similarly, with linear algebra, a norm, a scalar product, or a metric on a function space is often introduced. Some methods for solving differential equations , for example, the finite element method , are based on the concept of a function as an element of a Hilbert space .

General definition

The most general definition of a vector is given by means of general algebra :

We denote ${\mathfrak {F}}$ ${\ displaystyle {\ mathfrak {F}}}$ (Gothic F) some field with many elements $F$ ${\ displaystyle F}$ additive operation $+$ ${\ displaystyle +}$ , multiplicative operation $*$ ${\ displaystyle *}$ , and corresponding neutral elements : additive unit $0$ ${\ displaystyle 0}$ and multiplicative unit $one$ ${\ displaystyle 1}$ .
We denote ${\mathfrak {V}}$ ${\ displaystyle {\ mathfrak {V}}}$ (Gothic V) some abelian group with many elements $V$ ${\ displaystyle V}$ additive operation $+$ ${\ displaystyle +}$ and, accordingly, with the additive unit $\mathbf {0}$ ${\ displaystyle \ mathbf {0}}$ .

In other words, let ${\mathfrak {F}}=\langle F;+,*\rangle$ ${\ displaystyle {\ mathfrak {F}} = \ langle F; +, * \ rangle}$ and ${\mathfrak {V}}=\langle V;+\rangle$ ${\ displaystyle {\ mathfrak {V}} = \ langle V; + \ rangle}$ .

If an operation exists $F\times V\to V$ ${\ displaystyle F \ times V \ to V}$ such that for any $a,b\in F$ ${\ displaystyle a, b \ in F}$ and for any $\mathbf {x} ,\mathbf {y} \in V$ ${\ displaystyle \ mathbf {x}, \ mathbf {y} \ in V}$ the relations are satisfied:

$(a+b)\times \mathbf {x} =a\times \mathbf {x} +b\times \mathbf {x}$ ${\ displaystyle (a + b) \ times \ mathbf {x} = a \ times \ mathbf {x} + b \ times \ mathbf {x}}$ ,
$a\times (\mathbf {x} +\mathbf {y} )=a\times \mathbf {x} +a\times \mathbf {y}$ ${\ displaystyle a \ times (\ mathbf {x} + \ mathbf {y}) = a \ times \ mathbf {x} + a \ times \ mathbf {y}}$ ,
$(a*b)\times \mathbf {x} =a\times (b\times \mathbf {x} )$ ${\ displaystyle (a * b) \ times \ mathbf {x} = a \ times (b \ times \ mathbf {x})}$ ,
$1\times \mathbf {x} =\mathbf {x}$ ${\ displaystyle 1 \ times \ mathbf {x} = \ mathbf {x}}$ ,

then

${\mathfrak {V}}$ ${\ displaystyle {\ mathfrak {V}}}$ called the vector space over the field ${\mathfrak {F}}$ ${\ displaystyle {\ mathfrak {F}}}$ (or linear space)
the elements $V$ ${\ displaystyle V}$ called vectors ,
the elements $F$ ${\ displaystyle F}$ - scalars
specified operation $F\times V\to V$ ${\ displaystyle F \ times V \ to V}$ - multiplying the vector by a scalar .

Many results of linear algebra are generalized to unitary modules over non-commutative bodies and even arbitrary modules over rings , so in the most general case, in some contexts, any element of a module over a ring can be called a vector.

Physical Interpretation

Vector, as a structure having both magnitude (module) and direction, is considered in physics as a mathematical model of speed , force , and related quantities, kinematic or dynamic. The mathematical model of many physical fields (for example, electromagnetic fields or fluid velocity fields) are vector fields .

Abstract multidimensional and infinite-dimensional (in the spirit of functional analysis ) vector spaces are used in the Lagrangian and Hamiltonian formalism as applied to mechanical and other dynamical systems, as well as in quantum mechanics (see State Vector ).

Vector as a sequence

Vector - ( sequence , tuple ) of homogeneous elements. This is the most general definition in the sense that ordinary vector operations may not be specified at all, they may be less, or they may not satisfy the usual axioms of linear space . It is in this form that the vector is understood in programming , where, as a rule, it is indicated by the identifier name with square brackets (for example, object [] ). The list of properties models the definition of the class and state of an object accepted in system theory . So the types of vector elements determine the class of the object, and the values of the elements determine its state. However, it is likely that this use of the term already goes beyond what is usually accepted in algebra, and in mathematics in general.

An arithmetic vector is an ordered collection of n numbers. Designated ${\overline {x}}=(x_{1},x_{2},\ldots ,x_{n})$ ${\ displaystyle {\ overline {x}} = (x_ {1}, x_ {2}, \ ldots, x_ {n})}$ , numbers $x_{1},x_{2},\ldots ,x_{n}$ ${\ displaystyle x_ {1}, x_ {2}, \ ldots, x_ {n}}$ are called components of an arithmetic vector. The set of arithmetic vectors for which the operations of addition and multiplication by a number are defined is called the space of arithmetic vectors $R^{n}$ ${\ displaystyle R ^ {n}}$ . ^[3]

Notes

↑ Vector // Mathematical Encyclopedia (in 5 volumes) . - M .: Soviet Encyclopedia , 1977 .-- T. 1.
↑ Aleksandrova N.V. History of mathematical terms, concepts, notation: Dictionary-reference book. - 3rd ed. - SPb. : LCI, 2008 .-- pp. 22-23. - 248 p. - ISBN 978-5-382-00839-4 .
↑ Linear Algebra. IET MEI Short lecture notes http://old.exponenta.ru/educat/systemat/slivina/lection/lection3/lection3.asp

Literature

Gusyatnikov P.B., Reznichenko S.V. Vector algebra in examples and problems . - M .: Higher school , 1985 .-- 232 p.
Coxeter G. S. M. , Greitzer S. P. New Encounters with Geometry . - M .: Nauka , 1978. - T. 14. - ( Library of the mathematical circle ).
JV Field, The Invention of Infinity: Mathematics and Art in the Renaissance , Oxford University Press , 1997 ISBN 0198523947
F. Casiro, A. Deledicq, Pythagore et Thalès Les éditions du Kangourou 1998 ISBN 2-87694-040-X
R. Pouzergues, Les Hexamys , IREM de Nice, Irem Ouvrage, 1993 Cote: IM8974 Lire
D. Lehmann et Rudolf Bkouche , Initiation à la géométrie , PUF, 1988, ISBN 2130401600
Y. Sortais, La Géométrie du triangle. Exercices résolus , Hermann, 1997, ISBN 270561429X
Y. Ladegaillerie, Géométrie pour le CAPES de mathématiques , Ellipses Marketing, 2002 ISBN 2729811486
J. Perez, Mécanique physique , Masson, 2007 ISBN 2225553416
MB Karbo, Le graphisme et l'internet , Compétence micro, No. 26, 2002 ISBN 2912954959

Links

Premières utilisations connues des termes mathématiques par J. Miller

[1] Vector // Mathematical Encyclopedia (in 5 volumes) . - M .: Soviet Encyclopedia , 1977 .-- T. 1.

[2] Aleksandrova N.V. History of mathematical terms, concepts, notation: Dictionary-reference book. - 3rd ed. - SPb. : LCI, 2008 .-- pp. 22-23. - 248 p. - ISBN 978-5-382-00839-4 .

[3] Linear Algebra. IET MEI Short lecture notes http://old.exponenta.ru/educat/systemat/slivina/lection/lection3/lection3.asp