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Orthogonal Projection
In linear algebra and functional analysis , a PROJECTION is a linear transformation P from a vector space to itself such that P 2 = P. That is, whenever P is applied twice to any value, it gives the same result as if it were applied once (idempotent ). It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection . One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object
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Unit Vector
In mathematics , a UNIT VECTOR in a normed vector space is a vector (often a spatial vector ) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex , or "hat": {displaystyle {hat {imath }}} (pronounced "i-hat"). The term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as D. Two 2D direction vectors, D1 and D2 are illustrated. 2D spatial directions represented this way are numerically equivalent to points on the unit circle . The same construct is used to specify spatial directions in 3D. As illustrated, each unique direction is equivalent numerically to a point on the unit sphere
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Standard Inner Product
In mathematics , the DOT PRODUCT or SCALAR PRODUCT is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors ) and returns a single number. In Euclidean geometry , the dot product of the Cartesian coordinates of two vectors is widely used and often called INNER PRODUCT (or rarely PROJECTION PRODUCT); see also inner product space . Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry , Euclidean spaces are often defined by using vector spaces
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Outer Product
In linear algebra , an OUTER PRODUCT is the tensor product of two coordinate vectors , a special case of the Kronecker product of matrices. The outer product of two coordinate vectors u {displaystyle mathbf {u} } and v {displaystyle mathbf {v} } , denoted u v {displaystyle mathbf {u} otimes mathbf {v} } , is a matrix w {displaystyle mathbf {w} } such that the coordinates satisfy w i j = u i v j {displaystyle w_{ij}=u_{i}v_{j}} . The outer product for general tensors is also called the tensor product . The outer product contrasts with the dot product , which takes as input a pair of coordinate vectors and produces a scalar . The outer product is also a related function in some computer programming languages
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Dot Product
In mathematics , the DOT PRODUCT or SCALAR PRODUCT is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors ) and returns a single number. In Euclidean geometry , the dot product of the Cartesian coordinates of two vectors is widely used and often called INNER PRODUCT (or rarely PROJECTION PRODUCT); see also inner product space . Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry , Euclidean spaces are often defined by using vector spaces
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Orthonormal Basis
In mathematics , particularly linear algebra , an ORTHONORMAL BASIS for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal , that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation ) is also orthonormal, and every orthonormal basis for Rn arises in this fashion. For a general inner product space V, an orthonormal basis can be used to define normalized orthogonal coordinates on V. Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of Rn under dot product
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Cauchy–Schwarz Inequality
In mathematics , the CAUCHY–SCHWARZ INEQUALITY, also known as the CAUCHY–BUNYAKOVSKY–SCHWARZ INEQUALITY, is a useful inequality encountered in many different settings, such as linear algebra , analysis , probability theory , vector algebra and other areas. It is considered to be one of the most important inequalities in all of mathematics. It has a number of generalizations, among them Hölder\'s inequality . The inequality for sums was published by Augustin-Louis Cauchy
Augustin-Louis Cauchy
(1821 ), while the corresponding inequality for integrals was first proved by Viktor Bunyakovsky (1859 ). The modern proof of the integral inequality was given by Hermann Amandus Schwarz (1888 )
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Bounded Operator
In functional analysis , a branch of mathematics , a BOUNDED LINEAR OPERATOR is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in X. In other words, there exists some M > 0 such that for all v in X L v Y M v X . {displaystyle Lv_{Y}leq Mv_{X}.,,} The smallest such M is called the operator norm L o p {displaystyle L_{mathrm {op} },} of L. A bounded linear operator is generally not a bounded function ; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless Y is the zero vector space. Rather, a bounded linear operator is a locally bounded function . A linear operator between normed spaces is bounded if and only if it is continuous , and by linearity, if and only if it is continuous at zero
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Normed Vector Space
In mathematics , a NORMED VECTOR SPACE is a vector space on which a NORM is defined. In a vector space with 1- 2- or 3-dimensional vectors with real -valued entries, the idea of the "length" of a vector is intuitive. This intuition can easily be extended to any real vector space R n {displaystyle mathbb {R} ^{n}} . The length of a vector in such a vector space has the following properties: * The zero vector, 0, has zero length; every other vector has a positive length. x 0 {displaystyle xgeq 0} , and x = 0 {displaystyle x=0} if and only if x = 0 {displaystyle x=0} * Multiplying a vector by a positive number changes its length without changing its direction. Moreover, x = x {displaystyle alpha x=alpha x} for any scalar . {displaystyle alpha .} * The triangle inequality holds
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Infimum
In mathematics , the INFIMUM (abbreviated INF; plural INFIMA) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used. The SUPREMUM (abbreviated SUP; plural SUPREMA) of a subset S of a partially ordered set T is the least element in T that is greater than or equal to all elements of S, if such an element exists. Consequently, the supremum is also referred to as the least upper bound (or LUB). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis , and especially in Lebesgue integration . However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered
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Minimum
In mathematical analysis , the MAXIMA AND MINIMA (the respective plurals of MAXIMUM and MINIMUM) of a function , known collectively as EXTREMA (the plural of EXTREMUM), are the largest and smallest value of the function, either within a given range (the LOCAL or RELATIVE extrema) or on the entire domain of a function (the GLOBAL or ABSOLUTE extrema). Pierre de Fermat
Pierre de Fermat
was one of the first mathematicians to propose a general technique, adequality , for finding the maxima and minima of functions. As defined in set theory , the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers , have no minimum or maximum
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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 . In linear algebra , real numbers or other elements of a field are called SCALARS and relate to vectors in a vector space through the operation of scalar multiplication , in which a vector can be multiplied by a number to produce another 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 vector 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, allowing two vectors to be multiplied to produce a scalar. 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
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Partial Isometry
In functional analysis a PARTIAL ISOMETRY is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel . The orthogonal complement of its kernel is called the INITIAL SUBSPACE and its range is called the FINAL SUBSPACE. Partial isometries appear in the polar decomposition . CONTENTS * 1 General * 2 Operator Algebras * 3 C*-Algebras * 4 Special Classes * 4.1 Projections * 4.2 Embeddings * 4.3 Unitaries * 5 Examples * 5.1 Nilpotents * 5.2 Leftshift and Rightshift * 6 References * 7 External links GENERALThe concept of partial isometry can be defined in other equivalent ways. If U is an isometric map defined on a closed subset H1 of a Hilbert space H then we can define an extension W of U to all of H by the condition that W be zero on the orthogonal complement of H1. Thus a partial isometry is also sometimes defined as a closed partially defined isometric map
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Frame Of A Vector Space
In linear algebra , a FRAME of an inner product space is a generalization of a basis of a vector space to sets that may be linearly dependent . In the terminology of signal processing , a frame provides a redundant, stable way of representing a signal . Frames are used in error detection and correction and the design and analysis of filter banks and more generally in applied mathematics , computer science , and engineering
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Closed Graph Theorem
In mathematics , the CLOSED GRAPH THEOREM is a basic result which characterizes continuous functions in terms of their graphs . There are several versions of the theorem. CONTENTS* 1 The closed graph theorem * 1.1 In point-set topology * 1.2 In functional analysis * 2 Generalization * 2.1 The Borel Graph Theorem * 3 See also * 4 Notes * 5 References THE CLOSED GRAPH THEOREMIn mathematics, there are several results known as the "closed graph theorem". IN POINT-SET TOPOLOGYFor any function T : X → Y, we define the graph of T to be the set { ( x , y ) X Y T x = y }
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Bounded Linear Operator
In functional analysis , a branch of mathematics , a BOUNDED LINEAR OPERATOR is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded by the same number, over all non-zero vectors v in X. In other words, there exists some M > 0 such that for all v in X L v Y M v X . {displaystyle Lv_{Y}leq Mv_{X}.,,} The smallest such M is called the operator norm L o p {displaystyle L_{mathrm {op} },} of L. A bounded linear operator is generally not a bounded function ; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless L(v)=0 for all v. Rather, a bounded linear operator is a locally bounded function . A linear operator between normed spaces is bounded if and only if it is continuous , and by linearity, if and only if it is continuous at zero
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