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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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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 , 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, but these are better known as multiview projections
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Vector Projection
The VECTOR PROJECTION of a vector A on (or onto) a nonzero vector B (also known as the VECTOR COMPONENT or VECTOR RESOLUTION of A in the direction of B) is the orthogonal projection of A onto a straight line parallel to B. It is a vector parallel to B, defined as a 1 = a 1 b {displaystyle mathbf {a} _{1}=a_{1}mathbf {hat {b}} ,} where a 1 {displaystyle a_{1}} is a scalar, called the scalar projection of A onto B, and B̂ is the unit vector in the direction of B. In turn, the scalar projection is defined as a 1 = a cos = a b = a b b {displaystyle a_{1}=mathbf {a} cos theta =mathbf {a} cdot mathbf {hat {b}} =mathbf {a} cdot {frac {mathbf {b} }{mathbf {b} }},} where the operator · denotes a dot product , A is the length of A, and θ is the angle between A and B
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Linear Algebra
LINEAR ALGEBRA is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces. The set of points with coordinates that satisfy a linear equation forms a hyperplane in an _n_-dimensional space . The conditions under which a set of _n_ hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors. Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces
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Functional Analysis
FUNCTIONAL ANALYSIS is a branch of mathematical analysis , the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product , norm , topology , etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous , unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations . The usage of the word _functional _ goes back to the calculus of variations , implying a function whose argument is a function and the name was first used in Hadamard 's 1910 book on that subject
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Linear Transformation
In mathematics , a LINEAR MAP (also called a LINEAR MAPPING, LINEAR TRANSFORMATION or, in some contexts, LINEAR FUNCTION) is a mapping V → W between two modules (including vector spaces ) that preserves (in the sense defined below) the operations of addition and scalar multiplication. An important special case is when V = W, in which case the map is called a LINEAR OPERATOR, or an endomorphism of V. Sometimes the term linear function has the same meaning as linear map, while in analytic geometry it does not. A linear map always maps linear subspaces onto linear subspaces (possibly of a lower dimension); for instance it maps a plane through the origin to a plane , straight line or point . Linear maps can often be represented as matrices , and simple examples include rotation and reflection linear transformations . In the language of abstract algebra , a linear map is a module homomorphism
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Vector Space
A VECTOR SPACE (also called a LINEAR SPACE) is a collection of objects called VECTORS, which may be added together and multiplied ("scaled") by numbers, called scalars . Scalars are often taken to be real numbers , but there are also vector spaces with scalar multiplication by complex numbers , rational numbers , or generally any field . The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms , listed below . Euclidean vectors are an example of a vector space. They represent physical quantities such as forces : any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector
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Idempotence
IDEMPOTENCE (UK: /ˌɪdɛmˈpoʊtns/ ; US: /ˌaɪdəmˈpoʊtəns/ EYE-dəm-POH-təns ) is the property of certain operations in mathematics and computer science , that can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators ) and functional programming (in which it is connected to the property of referential transparency ). The term was introduced by Benjamin Peirce in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from idem + potence (same + power)
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Graphical Projection
GRAPHICAL PROJECTION is a protocol, used in technical drawing , by which an image of a three-dimensional object is projected onto a planar surface without the aid of numerical calculation
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Geometry
GEOMETRY (from the Ancient Greek : γεωμετρία; _geo-_ "earth", _-metron_ "measurement") 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 formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid , whose treatment, Euclid\'s _Elements_ , set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC
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Point (geometry)
In modern mathematics , a POINT refers usually to an element of some set called a space . More specifically, in Euclidean geometry , a point is a primitive notion upon which the geometry is built, meaning that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms , that it must satisfy. In particular, the geometric points do not have any length , area , volume or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space
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Matrix (mathematics)
In mathematics , a MATRIX (plural: MATRICES) is a rectangular array of numbers , symbols , or expressions , arranged in rows and columns. For example, the dimensions of the matrix below are 2 × 3 (read "two by three"), because there are two rows and three columns: . {displaystyle {begin{bmatrix}1&9&-13\20&5 width:16.213ex; height:6.176ex;" alt="{displaystyle {begin{bmatrix}1&9&-13\20&5 for instance, they are used within the PageRank
PageRank
algorithm that ranks the pages in a Google search. Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions. Matrices are used in economics to describe systems of economic relationships. A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research
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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 numerous applications in applied mathematics , physics , and engineering . In more detail, if A is an n × m matrix and B is an m × p matrix, their matrix product AB is an n × p matrix, in which the m entries across a row of A are multiplied with the m entries down a columns of B and summed to produce an entry of AB. When two linear transformations are represented by matrices, then the matrix product 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
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Linear Subspace
In linear algebra and related fields of mathematics , a LINEAR SUBSPACE, also known as a VECTOR SUBSPACE, or, in the older literature, a LINEAR MANIFOLD, is a vector space that is a subset of some other (higher-dimension ) vector space. A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces
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Range Of A Matrix
In linear algebra , the COLUMN SPACE (also called the RANGE or IMAGE ) of a matrix A is the span (set of all possible linear combinations ) of its column vectors . The column space of a matrix is the image or range of the corresponding matrix transformation . Let F {displaystyle mathbb {F} } be a field . The column space of an m × n matrix with components from F {displaystyle mathbb {F} } is a linear subspace of the m-space F m {displaystyle mathbb {F} ^{m}} . The dimension of the column space is called the rank of the matrix and is at most min(m, n). A definition for matrices over a ring K {displaystyle mathbb {K} } is also possible . The ROW SPACE is defined similarly. This article considers matrices of real numbers . The row and column spaces are subspaces of the real spaces Rn and Rm respectively
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Null Space
In mathematics , and more specifically in linear algebra and functional analysis , the KERNEL (also known as NULL SPACE or NULLSPACE) of a linear map L : V → W between two vector spaces V and W, is the set of all elements V of V for which L(V) = 0, where 0 denotes the zero vector in W. That is, in set-builder notation , ker ( L ) = { v V L ( v ) = 0 }
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