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Mathematics
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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

2.
Linear map
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In mathematics, a linear map is a mapping V → W between two modules that preserves the operations of addition and scalar multiplication. An important special case is when V = W, in case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the meaning as linear map. A linear map always maps linear subspaces onto linear subspaces, for instance it maps a plane through the origin to a plane, Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring, let V and W be vector spaces over the same field K. e. that for any vectors x1. Am ∈ K, the equality holds, f = a 1 f + ⋯ + a m f. It is then necessary to specify which of these fields is being used in the definition of linear. If V and W are considered as spaces over the field K as above, for example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear. A linear map from V to K is called a linear functional and these statements generalize to any left-module RM over a ring R without modification, and to any right-module upon reversing of the scalar multiplication. The zero map between two left-modules over the ring is always linear. The identity map on any module is a linear operator, any homothecy centered in the origin of a vector space, v ↦ c v where c is a scalar, is a linear operator. This does not hold in general for modules, where such a map might only be semilinear, for real numbers, the map x ↦ x2 is not linear. Conversely, any map between finite-dimensional vector spaces can be represented in this manner, see the following section. Differentiation defines a map from the space of all differentiable functions to the space of all functions. It also defines an operator on the space of all smooth functions. If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f, V → W to dimF × dimF matrices in the way described in the sequel are themselves linear maps. The expected value of a variable is linear, as for random variables X and Y we have E = E + E and E = aE

3.
Topological vector space
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In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure with the concept of a vector space. Hilbert spaces and Banach spaces are well-known examples, unless stated otherwise, the underlying field of a topological vector space is assumed to be either the complex numbers C or the real numbers R. Some authors require the topology on X to be T1, it follows that the space is Hausdorff. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the category of topological vector spaces over a given topological field K is commonly denoted TVSK or TVectK. The objects are the vector spaces over K and the morphisms are the continuous K-linear maps from one object to another. Every normed vector space has a topological structure, the norm induces a metric. This is a vector space because, The vector addition +, V × V → V is jointly continuous with respect to this topology. This follows directly from the triangle inequality obeyed by the norm, the scalar multiplication ·, K × V → V, where K is the underlying scalar field of V, is jointly continuous. This follows from the inequality and homogeneity of the norm. Therefore, all Banach spaces and Hilbert spaces are examples of vector spaces. There are topological spaces whose topology is not induced by a norm. These are all examples of Montel spaces, an infinite-dimensional Montel space is never normable. A topological field is a vector space over each of its subfields. A cartesian product of a family of vector spaces, when endowed with the product topology, is a topological vector space. For instance, the set X of all functions f, R → R, with this topology, X becomes a topological vector space, called the space of pointwise convergence. The reason for this name is the following, if is a sequence of elements in X, then fn has limit f in X if and only if fn has limit f for every real number x. This space is complete, but not normable, indeed, every neighborhood of 0 in the topology contains lines

4.
Linear form
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In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. The set of all linear functionals from V to k, Homk, forms a space over k with the addition of the operations of addition. This space is called the space of V, or sometimes the algebraic dual space. It is often written V∗ or V′ when the field k is understood, if V is a topological vector space, the space of continuous linear functionals — the continuous dual — is often simply called the dual space. If V is a Banach space, then so is its dual, to distinguish the ordinary dual space from the continuous dual space, the former is sometimes called the algebraic dual. In finite dimensions, every linear functional is continuous, so the dual is the same as the algebraic dual. Suppose that vectors in the coordinate space Rn are represented as column vectors x → =. For each row there is a linear functional f defined by f = a 1 x 1 + ⋯ + a n x n. This is just the product of the row vector and the column vector x →, f =. Linear functionals first appeared in functional analysis, the study of spaces of functions. Let Pn denote the space of real-valued polynomial functions of degree ≤n defined on an interval. If c ∈, then let evc, Pn → R be the evaluation functional, the mapping f → f is linear since = f + g = α f. If x0, …, xn are n+1 distinct points in, then the evaluation functionals evxi, the integration functional I defined above defines a linear functional on the subspace Pn of polynomials of degree ≤ n. If x0, …, xn are n+1 distinct points in, then there are coefficients a0, … and this forms the foundation of the theory of numerical quadrature. This follows from the fact that the linear functionals evxi, f → f defined above form a basis of the space of Pn. Linear functionals are particularly important in quantum mechanics, quantum mechanical systems are represented by Hilbert spaces, which are anti–isomorphic to their own dual spaces. A state of a mechanical system can be identified with a linear functional. For more information see bra–ket notation, in the theory of generalized functions, certain kinds of generalized functions called distributions can be realized as linear functionals on spaces of test functions

5.
Kernel (linear algebra)
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That is, in set-builder notation, ker =. The kernel of L is a subspace of the domain V. In the linear map L, V → W, two elements of V have the image in W if and only if their difference lies in the kernel of L, L = L ⇔ L =0. It follows that the image of L is isomorphic to the quotient of V by the kernel and this implies the rank–nullity theorem, dim + dim = dim . Where, by “rank” we mean the dimension of the image of L, when V is an inner product space, the quotient V / ker can be identified with the orthogonal complement in V of ker. This is the generalization to linear operators of the row space, or coimage, the notion of kernel applies to the homomorphisms of modules, the latter being a generalization of the vector space over a field to that over a ring. The domain of the mapping is a module, and the kernel constitutes a submodule, here, the concepts of rank and nullity do not necessarily apply. If V and W are topological vector spaces then a linear operator L, V → W is continuous if, consider a linear map represented as a m × n matrix A with coefficients in a field K and operating on column vectors x with n components over K. The kernel of this map is the set of solutions to the equation A x =0. The dimension of the kernel of A is called the nullity of A, in set-builder notation, N = Null = ker =. Thus the kernel of A is the same as the set to the above homogeneous equations. The kernel of an m × n matrix A over a field K is a subspace of Kn. That is, the kernel of A, the set Null, has the three properties, Null always contains the zero vector, since A0 =0. If x ∈ Null and y ∈ Null, then x + y ∈ Null and this follows from the distributivity of matrix multiplication over addition. If x ∈ Null and c is a scalar c ∈ K, then cx ∈ Null, the product Ax can be written in terms of the dot product of vectors as follows, A x =. , am denote the transposed rows of the matrix A and it follows that x is in the kernel of A if and only if x is orthogonal to each of the row vectors of A. The row space, or coimage, of a matrix A is the span of the row vectors of A, by the above reasoning, the kernel of A is the orthogonal complement to the row space. That is, a vector x lies in the kernel of A if, the dimension of the row space of A is called the rank of A, and the dimension of the kernel of A is called the nullity of A