Mathematical objectA mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an object is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs. Typically, a mathematical object can be a value that can be assigned to a variable, and therefore can be involved in formulas. Commonly encountered mathematical objects include numbers, sets, functions, expressions, geometric objects, transformations of other mathematical objects, and spaces.
Projection (linear algebra)In linear algebra and functional analysis, a projection is a linear transformation from a vector space to itself (an endomorphism) such that . That is, whenever is applied twice to any vector, it gives the same result as if it were applied once (i.e. is idempotent). It leaves its unchanged. 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.
Polar decompositionIn mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form , where is a unitary matrix and is a positive semi-definite Hermitian matrix ( is an orthogonal matrix and is a positive semi-definite symmetric matrix in the real case), both square and of the same size. Intuitively, if a real matrix is interpreted as a linear transformation of -dimensional space , the polar decomposition separates it into a rotation or reflection of , and a scaling of the space along a set of orthogonal axes.
Orthogonal complementIn the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W⊥ of all vectors in V that are orthogonal to every vector in W. Informally, it is called the perp, short for perpendicular complement. It is a subspace of V. Let be the vector space equipped with the usual dot product (thus making it an inner product space), and let with then its orthogonal complement can also be defined as being The fact that every column vector in is orthogonal to every column vector in can be checked by direct computation.
FormulaIn science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a chemical formula. The informal use of the term formula in science refers to the general construct of a relationship between given quantities. The plural of formula can be either formulas (from the most common English plural noun form) or, under the influence of scientific Latin, formulae (from the original Latin). In mathematics, a formula generally refers to an equation relating one mathematical expression to another, with the most important ones being mathematical theorems.
Binary relationIn mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. A binary relation over sets X and Y is a new set of ordered pairs (x, y) consisting of elements x in X and y in Y. It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element x is related to an element y, if and only if the pair (x, y) belongs to the set of ordered pairs that defines the binary relation.
Converse relationIn mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if and are sets and is a relation from to then is the relation defined so that if and only if In set-builder notation, The notation is analogous with that for an inverse function. Although many functions do not have an inverse, every relation does have a unique converse.
Axonometric projectionAxonometric projection is a type of orthographic projection used for creating a pictorial drawing of an object, where the object is rotated around one or more of its axes to reveal multiple sides. "Axonometry" means "to measure along the axes". In German literature, axonometry is based on Pohlke's theorem, such that the scope of axonometric projection could encompass every type of parallel projection, including not only orthographic projection (and multiview projection), but also oblique projection.
Rotations and reflections in two dimensionsIn Euclidean geometry, two-dimensional rotations and reflections are two kinds of Euclidean plane isometries which are related to one another. A rotation in the plane can be formed by composing a pair of reflections. First reflect a point P to its image P′ on the other side of line L1. Then reflect P′ to its image P′′ on the other side of line L2. If lines L1 and L2 make an angle θ with one another, then points P and P′′ will make an angle 2θ around point O, the intersection of L1 and L2. I.e.
Coordinate systemIn geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring.
Homogeneous relationIn mathematics, a homogeneous relation (also called endorelation) on a set X is a binary relation between X and itself, i.e. it is a subset of the Cartesian product X × X. This is commonly phrased as "a relation on X" or "a (binary) relation over X". An example of a homogeneous relation is the relation of kinship, where the relation is between people. Common types of endorelations include orders, graphs, and equivalences. Specialized studies of order theory and graph theory have developed understanding of endorelations.
Reflexive relationIn mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself. An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.
