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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.[1] Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection
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Orthographic Projection
Orthographic projection
Orthographic projection
(sometimes orthogonal projection), is a means of representing three-dimensional objects in two dimensions. It is a form of parallel projection, in which all the projection lines are orthogonal to the projection plane,[1] resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are not orthogonal to the projection plane. The term orthographic is sometimes reserved specifically for depictions of objects where the principal axes or planes of the object are also parallel with the projection plane,[1] but these are better known as multiview projections
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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
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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).[1][2][3] 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
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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.[1] 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.[2][3][4] 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
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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 above 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
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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.[1] The inequality for sums was published by Augustin-Louis Cauchy (1821), while the corresponding inequality for integrals was first proved by Viktor Bunyakovsky (1859)
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Standard Inner Product
In mathematics, the dot product or scalar product[note 1] 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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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": ı ^ 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
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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
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Dot Product
In mathematics, the dot product or scalar product[note 1] 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
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.[1][2][3] For example, the standard basis for a Euclidean space
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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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.Contents1 General 2 Operator Algebras 3 C*-Algebras 4 Special
Special
Classes4.1 Projections 4.2 Embeddings 4.3 Unitaries5 Examples5.1 Nilpotents 5.2 Leftshift and Rightshift6 References 7 External linksGeneral[edit] The 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
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Moore–Penrose Pseudoinverse
In mathematics, and in particular linear algebra, a pseudoinverse A+ of a matrix A is a generalization of the inverse matrix.[1] The most widely known type of matrix pseudoinverse is the Moore–Penrose inverse,[2][3][4][5] which was independently described by E. H. Moore[6] in 1920, Arne Bjerhammar[7] in 1951, and Roger Penrose[8] in 1955. Earlier, Erik Ivar Fredholm
Erik Ivar Fredholm
had introduced the concept of a pseudoinverse of integral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose inverse. The term generalized inverse is sometimes used as a synonym for pseudoinverse. A common use of the pseudoinverse is to compute a 'best fit' (least squares) solution to a system of linear equations that lacks a unique solution (see below under § Applications)
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Normed Vector Space
In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of distance in the real world. A norm is a real-valued function defined on the vector space that 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
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Conjugate Transpose
In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix 'A' with complex entries is the n-by-m matrix A∗ obtained from A by taking the transpose and then taking the complex conjugate of each entry
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