1.
Inner product space
–
In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a quantity known as the inner product of the vectors. Inner products allow the introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors, inner product spaces generalize Euclidean spaces to vector spaces of any dimension, and are studied in functional analysis. An inner product induces a associated norm, thus an inner product space is also a normed vector space. A complete space with a product is called a Hilbert space. An space with a product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space. Inner product spaces over the field of numbers are sometimes referred to as unitary spaces. In this article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C, formally, an inner product space is a vector space V over the field F together with an inner product, i. e. Some authors, especially in physics and matrix algebra, prefer to define the inner product, then the first argument becomes conjugate linear, rather than the second. In those disciplines we would write the product ⟨ x, y ⟩ as ⟨ y | x ⟩, respectively y † x. Here the kets and columns are identified with the vectors of V and this reverse order is now occasionally followed in the more abstract literature, taking ⟨ x, y ⟩ to be conjugate linear in x rather than y. A few instead find a ground by recognizing both ⟨ ⋅, ⋅ ⟩ and ⟨ ⋅ | ⋅ ⟩ as distinct notations differing only in which argument is conjugate linear. There are various reasons why it is necessary to restrict the basefield to R and C in the definition. Briefly, the basefield has to contain an ordered subfield in order for non-negativity to make sense, the basefield has to have additional structure, such as a distinguished automorphism. More generally any quadratically closed subfield of R or C will suffice for this purpose, however in these cases when it is a proper subfield even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over R or C, such as used in quantum computation, are automatically metrically complete. In some cases we need to consider non-negative semi-definite sesquilinear forms and this means that ⟨ x, x ⟩ is only required to be non-negative
2.
Scalar multiplication
–
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra. In common geometrical contexts, scalar multiplication of a real Euclidean vector by a real number multiplies the magnitude of the vector without changing its direction. The term scalar itself derives from this usage, a scalar is that which scales vectors, scalar multiplication is the multiplication of a vector by a scalar, and must be distinguished from inner product of two vectors. In general, if K is a field and V is a space over K. The result of applying this function to c in K and v in V is denoted cv, here + is addition either in the field or in the vector space, as appropriate, and 0 is the additive identity in either. Juxtaposition indicates either scalar multiplication or the operation in the field. Scalar multiplication may be viewed as a binary operation or as an action of the field on the vector space. A geometric interpretation of scalar multiplication is that it stretches, or contracts, as a special case, V may be taken to be K itself and scalar multiplication may then be taken to be simply the multiplication in the field. When V is Kn, scalar multiplication is equivalent to multiplication of each component with the scalar, the same idea applies if K is a commutative ring and V is a module over K. K can even be a rig, but then there is no additive inverse. If K is not commutative, the operations left scalar multiplication cv. The left scalar multiplication of a matrix A with a scalar λ gives another matrix λA of the size as A. The entries of λA are defined by i j = λ i j, explicitly, similarly, the right scalar multiplication of a matrix A with a scalar λ is defined to be i j = i j λ, explicitly, A λ = λ =. When the underlying ring is commutative, for example, the real or complex number field, however, for matrices over a more general ring that are not commutative, such as the quaternions, they may not be equal. For a real scalar and matrix, λ =2, A =2 A =2 = = =2 = A2. For quaternion scalars and matrices, λ = i, A = i = = ≠ = = i, the non-commutativity of quaternion multiplication prevents the transition of changing ij = +k to ji = −k
3.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
4.
Algebraic operation
–
These operations may be performed on numbers, in which case they are often called arithmetic operations. They may also be performed, in a way, on variables, algebraic expressions. The term algebraic operation may also be used for operations that may be defined by compounding basic algebraic operations, in calculus and mathematical analysis, algebraic operation is also used for the operations that may be defined by purely algebraic methods. For example, exponentiation with an integer or rational exponent is an algebraic operation, also, the derivative is an operation that is not algebraic. Multiplication symbols are usually omitted, and implied when there is no operator between two variables or terms, or when a coefficient is used, for example,3 × x2 is written as 3x2, and 2 × x × y is written as 2xy. Sometimes multiplication symbols are replaced with either a dot, or center-dot, plain text, programming languages, and calculators also use a single asterisk to represent the multiplication symbol, and it must be explicitly used, for example, 3x is written as 3 * x. Rather than using the symbol, ÷, division is usual represented with a vinculum. In plain text and programming languages a slash is used, e. g.3 /, exponents are usually formatted using superscripts, e. g. x2. In plain text, and in the TeX mark-up language, the symbol, ^, represents exponents. In programming languages such as Ada, Fortran, Perl, Python and Ruby, the plus-minus sign, ±, is used as a shorthand notation for two expressions written as one, representing one expression with a plus sign, the other with a minus sign. For example y = x ±1 represents the two equations y = x +1 and y = x −1, sometimes it is used for denoting positive-or-negative term such as ±x. Algebraic operations work in the way as arithmetic operations, as can be seen in the table below. Note, the use of the letters a and b is arbitrary, and the examples would be equally valid if we had used x and y
5.
Euclidean geometry
–
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry, the Elements. Euclids method consists in assuming a set of intuitively appealing axioms. Although many of Euclids results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in school as the first axiomatic system. It goes on to the geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, for more than two thousand years, the adjective Euclidean was unnecessary because no other sort of geometry had been conceived. Euclids axioms seemed so obvious that any theorem proved from them was deemed true in an absolute, often metaphysical. Today, however, many other self-consistent non-Euclidean geometries are known, Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates, the Elements is mainly a systematization of earlier knowledge of geometry. Its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones. There are 13 total books in the Elements, Books I–IV, Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved, a typical result is the 1,3 ratio between the volume of a cone and a cylinder with the same height and base. Euclidean geometry is a system, in which all theorems are derived from a small number of axioms. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. Although Euclids statement of the only explicitly asserts the existence of the constructions. The Elements also include the five common notions, Things that are equal to the same thing are also equal to one another
6.
Cartesian coordinates
–
Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
7.
Product (mathematics)
–
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. Thus, for instance,6 is the product of 2 and 3, the order in which real or complex numbers are multiplied has no bearing on the product, this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied, matrix multiplication, for example, and multiplication in other algebras is in general non-commutative. There are many different kinds of products in mathematics, besides being able to multiply just numbers, polynomials or matricies, an overview of these different kinds of products is given here. Placing several stones into a pattern with r rows and s columns gives r ⋅ s = ∑ i =1 s r = ∑ j =1 r s stones. Integers allow positive and negative numbers, the product of two quaternions can be found in the article on quaternions. However, it is interesting to note that in this case, the product operator for the product of a sequence is denoted by the capital Greek letter Pi ∏. The product of a sequence consisting of one number is just that number itself. The product of no factors at all is known as the empty product, commutative rings have a product operation. Under the Fourier transform, convolution becomes point-wise function multiplication, others have very different names but convey essentially the same idea. A brief overview of these is given here, by the very definition of a vector space, one can form the product of any scalar with any vector, giving a map R × V → V. A scalar product is a map, ⋅, V × V → R with the following conditions. From the scalar product, one can define a norm by letting ∥ v ∥, = v ⋅ v, now we consider the composition of two linear mappings between finite dimensional vector spaces. Let the linear mapping f map V to W, and let the linear mapping g map W to U, then one can get g ∘ f = g = g j k f i j v i b U k. Or in matrix form, g ∘ f = G F v, in which the i-row, j-column element of F, denoted by Fij, is fji, the composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication. To see this, let r = dim, s = dim, let U = be a basis of U, V = be a basis of V and W = be a basis of W. Then B ⋅ A = M W U ∈ R s × t is the matrix representing g ∘ f, U → W, in other words, the matrix product is the description in coordinates of the composition of linear functions. For inifinite-dimensional vector spaces, one also has the, Tensor product of Hilbert spaces Topological tensor product, the tensor product, outer product and Kronecker product all convey the same general idea
8.
Euclidean vector
–
In mathematics, physics, and engineering, a Euclidean vector is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra, a Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by A B →. A vector is what is needed to carry the point A to the point B and it was first used by 18th century astronomers investigating planet rotation around the Sun. The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space. Vectors play an important role in physics, the velocity and acceleration of a moving object, many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances, their magnitude and direction can still be represented by the length, the mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the system include pseudovectors and tensors. The concept of vector, as we know it today, evolved gradually over a period of more than 200 years, about a dozen people made significant contributions. Giusto Bellavitis abstracted the basic idea in 1835 when he established the concept of equipollence, working in a Euclidean plane, he made equipollent any pair of line segments of the same length and orientation. Essentially he realized an equivalence relation on the pairs of points in the plane, the term vector was introduced by William Rowan Hamilton as part of a quaternion, which is a sum q = s + v of a Real number s and a 3-dimensional vector. Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments, grassmanns work was largely neglected until the 1870s. Peter Guthrie Tait carried the standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator ∇, in 1878 Elements of Dynamic was published by William Kingdon Clifford. Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from the complete quaternion product and this approach made vector calculations available to engineers and others working in three dimensions and skeptical of the fourth. Josiah Willard Gibbs, who was exposed to quaternions through James Clerk Maxwells Treatise on Electricity and Magnetism, the first half of Gibbss Elements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis. In 1901 Edwin Bidwell Wilson published Vector Analysis, adapted from Gibbs lectures, in physics and engineering, a vector is typically regarded as a geometric entity characterized by a magnitude and a direction. It is formally defined as a line segment, or arrow
9.
Trigonometric functions
–
In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane, Angles are also formed by the intersection of two planes in Euclidean and other spaces. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is also used to designate the measure of an angle or of a rotation and this measure is the ratio of the length of a circular arc to its radius. In the case of an angle, the arc is centered at the vertex. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. The word angle comes from the Latin word angulus, meaning corner, cognate words are the Greek ἀγκύλος, meaning crooked, curved, both are connected with the Proto-Indo-European root *ank-, meaning to bend or bow. Euclid defines a plane angle as the inclination to each other, in a plane, according to Proclus an angle must be either a quality or a quantity, or a relationship. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle, lower case Roman letters are also used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples, in geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB, sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex. However, in geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant. Otherwise, a convention may be adopted so that ∠BAC always refers to the angle from B to C. Angles smaller than an angle are called acute angles. An angle equal to 1/4 turn is called a right angle, two lines that form a right angle are said to be normal, orthogonal, or perpendicular. Angles larger than an angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle, Angles larger than a straight angle but less than 1 turn are called reflex angles
10.
Geometry
–
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
11.
Euclidean space
–
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space
12.
Vector space
–
A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers. The operations of addition and scalar multiplication must satisfy certain requirements, called axioms. Euclidean vectors are an example of a vector space and they represent physical quantities such as forces, any two forces can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces and these vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are commonly used. This is particularly the case of Banach spaces and Hilbert spaces, historically, the first ideas leading to vector spaces can be traced back as far as the 17th centurys analytic geometry, matrices, systems of linear equations, and Euclidean vectors. Today, vector spaces are applied throughout mathematics, science and engineering, furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques, Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra. The concept of space will first be explained by describing two particular examples, The first example of a vector space consists of arrows in a fixed plane. This is used in physics to describe forces or velocities, given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w, when a is negative, av is defined as the arrow pointing in the opposite direction, instead. Such a pair is written as, the sum of two such pairs and multiplication of a pair with a number is defined as follows, + = and a =. The first example above reduces to one if the arrows are represented by the pair of Cartesian coordinates of their end points. A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below, elements of V are commonly called vectors. Elements of F are commonly called scalars, the second operation, called scalar multiplication takes any scalar a and any vector v and gives another vector av. In this article, vectors are represented in boldface to distinguish them from scalars
13.
Square root
–
In mathematics, a square root of a number a is a number y such that y2 = a, in other words, a number y whose square is a. For example,4 and −4 are square roots of 16 because 42 =2 =16, every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the square root of 9 is 3, denoted √9 =3. The term whose root is being considered is known as the radicand, the radicand is the number or expression underneath the radical sign, in this example 9. Every positive number a has two roots, √a, which is positive, and −√a, which is negative. Together, these two roots are denoted ± √a, although the principal square root of a positive number is only one of its two square roots, the designation the square root is often used to refer to the principal square root. For positive a, the square root can also be written in exponent notation. Square roots of numbers can be discussed within the framework of complex numbers. In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, a method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra. Aryabhata in the Aryabhatiya, has given a method for finding the root of numbers having many digits. It was known to the ancient Greeks that square roots of positive numbers that are not perfect squares are always irrational numbers, numbers not expressible as a ratio of two integers. This is the theorem Euclid X,9 almost certainly due to Theaetetus dating back to circa 380 BC, the particular case √2 is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. Mahāvīra, a 9th-century Indian mathematician, was the first to state that square roots of negative numbers do not exist, a symbol for square roots, written as an elaborate R, was invented by Regiomontanus. An R was also used for Radix to indicate square roots in Gerolamo Cardanos Ars Magna, according to historian of mathematics D. E. Smith, Aryabhatas method for finding the root was first introduced in Europe by Cataneo in 1546. According to Jeffrey A. Oaks, Arabs used the letter jīm/ĝīm, the letter jīm resembles the present square root shape. Its usage goes as far as the end of the century in the works of the Moroccan mathematician Ibn al-Yasamin. The symbol √ for the root was first used in print in 1525 in Christoph Rudolffs Coss
14.
Interpunct
–
It appears in a variety of uses in some modern languages and is present in Unicode as code point U+00B7 · Middle dot. The multiplication dot, whose glyphs are similar or identical to the interpunct, is a multiplication sign optionally used instead of the styled ×, the same sign is also used in vector multiplication to discriminate between the scalar product and the vector cross product or exterior product. As a multiplication operator, it is encountered in symbols for compound units such as the newton-meter. The multiplication dot is a separate Unicode character, but is often replaced by the interpunct or bullet. Various dictionaries use the interpunct to indicate syllabification within a word with multiple syllables, there is also a separate Unicode character, U+2027 ‧ hyphenation point. In British typography, the dot is an interpunct used as the formal decimal point. Its use is advocated by laws and by academic circles such as the Cambridge University History Faculty Style Guide and is mandated by some UK-based academic journals such as The Lancet, the space dot may still be used frequently in handwriting, however. In the Shavian alphabet, interpuncts replace capitalization as the marker of proper nouns, the dot is placed at the beginning of a word. The flown dot is used in Catalan between two Ls in cases where each belongs to a syllable, for example cel·la, cell. This distinguishes such geminate Ls, which are pronounced, from double L, in situations where the flown dot is unavailable, periods or hyphens are frequently used as substitutes, but this is tolerated rather than encouraged. Historically, medieval Catalan also used the symbol ⟨·⟩ as a marker for certain elisions, much like the modern apostrophe, there is no separate keyboard layout for Catalan, the flown dot can be typed using Shift-3 in the Spanish layout. It appears in Unicode as the letters ⟨Ŀ⟩ and ⟨ŀ⟩, similarly, the larger bullet may be seen but is discouraged on aesthetic grounds. The preferred Unicode representation is ⟨l·⟩, the partition sign is used in Chinese to mark divisions in transliterated foreign words, particularly names. This is properly a full-width punctuation mark, although sometimes narrower forms are substituted for aesthetic reasons, in particular, the regular interpunct is more commonly used as a computer input, although Chinese-language fonts typically render this as full width. When the Chinese text is romanized, the sign is simply replaced by a standard space or other appropriate punctuation. Thus, William Shakespeare is signified as 威廉·莎士比亞 or 威廉·莎士比亚, George W. Bush as 喬治·W·布殊 or 乔治·W·布什, titles and other translated words are not similarly marked, Genghis Khan and Elizabeth II are simply 成吉思汗 and 伊利沙伯二世 or 伊麗莎白二世 without a partition sign. The partition sign is used to separate book and chapter titles when they are mentioned consecutively, book first. It also appears in Swedish sinologist Bernhard Karlgrens works, where an interpunct is used to represent the stop in his reconstruction of medieval Chinese
15.
Scalar (mathematics)
–
A scalar is an element of a field which is used to define a vector space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector, more generally, a vector space may be defined by using any field instead of real numbers, such as complex numbers. Then the scalars of that space will be the elements of the associated field. A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, a vector space equipped with a scalar product is called an inner product space. The real component of a quaternion is also called its scalar part, the term is also sometimes used informally to mean a vector, matrix, tensor, or other usually compound value that is actually reduced to a single component. Thus, for example, the product of a 1×n matrix and an n×1 matrix, the term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix. The word scalar derives from the Latin word scalaris, a form of scala. The English word scale also comes from scala, according to a citation in the Oxford English Dictionary the first recorded usage of the term scalar in English came with W. R. A vector space is defined as a set of vectors, a set of scalars, and a multiplication operation that takes a scalar k. For example, in a space, the scalar multiplication k yields. In a function space, kƒ is the function x ↦ k, the scalars can be taken from any field, including the rational, algebraic, real, and complex numbers, as well as finite fields. According to a theorem of linear algebra, every vector space has a basis. It follows that every vector space over a scalar field K is isomorphic to a vector space where the coordinates are elements of K. For example, every vector space of dimension n is isomorphic to n-dimensional real space Rn. Alternatively, a vector space V can be equipped with a function that assigns to every vector v in V a scalar ||v||. By definition. If ||v|| is interpreted as the length of v, this operation can be described as scaling the length of v by k, a vector space equipped with a norm is called a normed vector space. The norm is defined to be an element of Vs scalar field K. Moreover, if V has dimension 2 or more, K must be closed under square root, as well as the four operations, thus the rational numbers Q are excluded
16.
Cross product
–
In mathematics and vector algebra, the cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. Given two linearly independent vectors a and b, the product, a × b, is a vector that is perpendicular to both a and b and therefore normal to the plane containing them. It has many applications in mathematics, physics, engineering, and it should not be confused with dot product. If two vectors have the direction or if either one has zero length, then their cross product is zero. The cross product is anticommutative and is distributive over addition, the space R3 together with the cross product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket. Like the dot product, it depends on the metric of Euclidean space, but if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions. If one adds the further requirement that the product be uniquely defined, the cross product of two vectors a and b is defined only in three-dimensional space and is denoted by a × b. In physics, sometimes the notation a ∧ b is used, if the vectors a and b are parallel, by the above formula, the cross product of a and b is the zero vector 0. Then, the n is coming out of the thumb. Using this rule implies that the cross-product is anti-commutative, i. e. b × a = −. By pointing the forefinger toward b first, and then pointing the finger toward a. Using the cross product requires the handedness of the system to be taken into account. If a left-handed coordinate system is used, the direction of the n is given by the left-hand rule. This, however, creates a problem because transforming from one arbitrary reference system to another, the problem is clarified by realizing that the cross product of two vectors is not a vector, but rather a pseudovector. See cross product and handedness for more detail, in 1881, Josiah Willard Gibbs, and independently Oliver Heaviside, introduced both the dot product and the cross product using a period and an x, respectively, to denote them. These alternative names are widely used in the literature. Both the cross notation and the cross product were possibly inspired by the fact that each scalar component of a × b is computed by multiplying non-corresponding components of a and b. Conversely, a dot product a ⋅ b involves multiplications between corresponding components of a and b, as explained below, the cross product can be expressed in the form of a determinant of a special 3 ×3 matrix
17.
Cartesian coordinate system
–
Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
18.
Real coordinate space
–
In mathematics, real coordinate space of n dimensions, written Rn is a coordinate space that allows several real variables to be treated as a single variable. With various numbers of dimensions, Rn is used in areas of pure and applied mathematics. With component-wise addition and scalar multiplication, it is the real vector space and is a frequently used representation of Euclidean n-space. Due to the fact, geometric metaphors are widely used for Rn, namely a plane for R2. For any natural n, the set Rn consists of all n-tuples of real numbers. It is called n-dimensional real space, for each n there exists only one Rn, the real n-space. Purely mathematical uses of Rn can be classified as follows. First, linear algebra studies its own properties under vector addition and linear transformations, the third use parametrizes geometric points with elements of Rn, it is common in analytic, differential and algebraic geometries. Rn, together with structures on it, is also extensively used in mathematical physics, dynamical systems theory, mathematical statistics. In applied mathematics, numerical analysis, and so on, arrays, sequences, Any function f of n real variables can be considered as a function on Rn. The use of the real n-space, instead of several variables considered separately, can simplify notation, consider, for n =2, a function composition of the following form, F = f, where functions g1 and g2 are continuous. If ∀x1 ∈ R , f is continuous ∀x2 ∈ R , f is continuous then F is not necessarily continuous, continuity is a stronger condition, the continuity of f in the natural R2 topology, also called multivariable continuity, which is sufficient for continuity of the composition F. The coordinate space Rn forms a vector space over the field of real numbers with the addition of the structure of linearity. The operations on Rn as a space are typically defined by x + y = α x =. The zero vector is given by 0 = and the inverse of the vector x is given by − x =. This structure is important because any n-dimensional real vector space is isomorphic to the vector space Rn, in standard matrix notation, each element of Rn is typically written as a column vector x = and sometimes as a row vector, x =. The coordinate space Rn may then be interpreted as the space of all n × 1 column vectors, or all 1 × n row vectors with the matrix operations of addition. Linear transformations from Rn to Rm may then be written as matrices which act on the elements of Rn via left multiplication and on elements of Rm via right multiplication
19.
Summation
–
In mathematics, summation is the addition of a sequence of numbers, the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a sum, prefix sum. The numbers to be summed may be integers, rational numbers, real numbers, besides numbers, other types of values can be added as well, vectors, matrices, polynomials and, in general, elements of any additive group. For finite sequences of elements, summation always produces a well-defined sum. The summation of a sequence of values is called a series. A value of such a series may often be defined by means of a limit, another notion involving limits of finite sums is integration. The summation of the sequence is an expression whose value is the sum of each of the members of the sequence, in the example,1 +2 +4 +2 =9. Addition is also commutative, so permuting the terms of a sequence does not change its sum. There is no notation for the summation of such explicit sequences. If, however, the terms of the sequence are given by a pattern, possibly of variable length. For the summation of the sequence of integers from 1 to 100. In this case, the reader can guess the pattern. However, for more complicated patterns, one needs to be precise about the used to find successive terms. Using this sigma notation the above summation is written as, ∑ i =1100 i, the value of this summation is 5050. It can be found without performing 99 additions, since it can be shown that ∑ i =1 n i = n 2 for all natural numbers n, more generally, formulae exist for many summations of terms following a regular pattern. By contrast, summation as discussed in this article is called definite summation, when it is necessary to clarify that numbers are added with their signs, the term algebraic sum is used. Mathematical notation uses a symbol that compactly represents summation of many terms, the summation symbol, ∑. The i = m under the symbol means that the index i starts out equal to m
20.
Three-dimensional space
–
Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the meaning of the term dimension. In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space, when n =3, the set of all such locations is called three-dimensional Euclidean space. It is commonly represented by the symbol ℝ3 and this serves as a three-parameter model of the physical universe in which all known matter exists. However, this space is one example of a large variety of spaces in three dimensions called 3-manifolds. Furthermore, in case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and breadth. In mathematics, analytic geometry describes every point in space by means of three coordinates. Three coordinate axes are given, each perpendicular to the two at the origin, the point at which they cross. They are usually labeled x, y, and z, below are images of the above-mentioned systems. Two distinct points determine a line. Three distinct points are either collinear or determine a unique plane, four distinct points can either be collinear, coplanar or determine the entire space. Two distinct lines can intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a plane, so skew lines are lines that do not meet. Two distinct planes can either meet in a line or are parallel. Three distinct planes, no pair of which are parallel, can meet in a common line. In the last case, the three lines of intersection of each pair of planes are mutually parallel, a line can lie in a given plane, intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line, a hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a space are the two-dimensional subspaces, that is
21.
Matrix multiplication
–
In mathematics, matrix multiplication or the matrix product is a binary operation that produces a matrix from two matrices. The definition is motivated by linear equations and linear transformations on vectors, which have applications in applied mathematics, physics. When two linear transformations are represented by matrices, then the matrix represents the composition of the two transformations. The matrix product is not commutative in general, although it is associative and is distributive over matrix addition, the identity element of the matrix product is the identity matrix, and a square matrix may have an inverse matrix. Determinant multiplicativity applies to the matrix product, the matrix product is also important for matrix groups, and the theory of group representations and irreps. Computing matrix products is both an operation in many numerical algorithms and potentially time consuming, making it one of the most well-studied problems in numerical computing. Various algorithms have been devised for computing C = AB, especially for large matrices, index notation is often the clearest way to express definitions, and is used as standard in the literature. The i, j entry of matrix A is indicated by ij or Aij, whereas a numerical label on a collection of matrices is subscripted only, e. g. A1, A2, assume two matrices are to be multiplied. M, and summing the results over k, i j = ∑ k =1 m A i k B k j. Thus the product AB is defined if the number of columns in A is equal to the number of rows in B. Each entry may be computed one at a time, sometimes, the summation convention is used as it is understood to sum over the repeated index k. To prevent any ambiguity, this convention will not be used in the article, usually the entries are numbers or expressions, but can even be matrices themselves. The matrix product can still be calculated exactly the same way, see below for details on how the matrix product can be calculated in terms of blocks taking the forms of rows and columns. The figure to the right illustrates diagrammatically the product of two matrices A and B, showing how each intersection in the matrix corresponds to a row of A. Note AB and BA are two different matrices, the first is a 1 ×1 matrix while the second is a 3 ×3 matrix, if A =, B =, their matrix product is, A B = =, however BA is not defined. The product of a square matrix multiplied by a column matrix arises naturally in algebra, for solving linear equations. By choosing a, b, c, p, q, r, u, v, w in A appropriately, A can represent a variety of such as rotations, scaling and reflections, shears. If A =, B =, their products are, A B = =
![{\displaystyle {\begin{aligned}\ [{\color {red}1,3,-5}]\cdot [{\color {blue}4,-2,-1}]&=({\color {red}1}\times {\color {blue}4})+({\color {red}3}\times {\color {blue}-2})+({\color {red}-5}\times {\color {blue}-1})\\&=4-6+5\\&=3\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be560d2c22a074c7711ae946954725d31ec77928)
![{\displaystyle {\begin{aligned}\mathbf {a} &=[a_{1},\dots ,a_{n}]=\sum _{i}a_{i}\mathbf {e} _{i}\\\mathbf {b} &=[b_{1},\dots ,b_{n}]=\sum _{i}b_{i}\mathbf {e} _{i}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b154ac2bb09512c81d917db83c273055c093571f)
