Linear subspaceIn mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces. If V is a vector space over a field K and if W is a subset of V, then W is a linear subspace of V if under the operations of V, W is a vector space over K.
ProjectorA projector or image projector is an optical device that projects an (or moving images) onto a surface, commonly a projection screen. Most projectors create an image by shining a light through a small transparent lens, but some newer types of projectors can project the image directly, by using lasers. A virtual retinal display, or retinal projector, is a projector that projects an image directly on the retina instead of using an external projection screen. The most common type of projector used today is called a video projector.
Mathematical proofA mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
Linear mapIn mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a .
Linear ALinear A is a writing system that was used by the Minoans of Crete from 1800 to 1450 BC to write the hypothesized Minoan language or languages. Linear A was the primary script used in palace and religious writings of the Minoan civilization. It was succeeded by Linear B, which was used by the Mycenaeans to write an early form of Greek. It was discovered by archaeologist Sir Arthur Evans. No texts in Linear A have yet been deciphered.
Vector spaceIn mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space.
Video projectorA video projector is an that receives a video signal and projects the corresponding image on a projection screen using a lens system. Video projectors use a very bright ultra-high-performance lamp (a special mercury arc lamp), Xenon arc lamp, LED or solid state blue, RB, RGB or remote fiber optic RGB lasers to provide the illumination required to project the image, and most modern ones can correct any curves, blurriness, and other inconsistencies through manual settings.
Movie projectorA movie projector is an opto-mechanical device for displaying motion picture film by projecting it onto a screen. Most of the optical and mechanical elements, except for the illumination and sound devices, are present in movie cameras. Modern movie projectors are specially built video projectors (see also digital cinema). Many projectors are specific to a particular film gauge and not all movie projectors are film projectors since the use of film is required. The main precursor to the movie projector was the magic lantern.
Linear BLinear B is a syllabic script that was used for writing in Mycenaean Greek, the earliest attested form of the Greek language. The script predates the Greek alphabet by several centuries. The oldest Mycenaean writing dates to about 1400 BC. It is descended from the older Linear A, an undeciphered earlier script used for writing the Minoan language, as is the later Cypriot syllabary, which also recorded Greek. Linear B, found mainly in the palace archives at Knossos, Kydonia, Pylos, Thebes and Mycenae, disappeared with the fall of Mycenaean civilization during the Late Bronze Age collapse.
Topological vector spaceIn mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar multiplication) are also continuous functions. Such a topology is called a and every topological vector space has a uniform topological structure, allowing a notion of uniform convergence and completeness.
Overhead projectorAn overhead projector (often abbreviated to OHP), like a film or slide projector, uses light to project an enlarged image on a screen, allowing the view of a small document or picture to be shared with a large audience. In the overhead projector, the source of the image is a page-sized sheet of transparent plastic film (also known as "foils" or "transparencies") with the image to be projected either printed or hand-written/drawn.
Normed vector spaceIn mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. A norm is a generalization of the intuitive notion of "length" in the physical world. If is a vector space over , where is a field equal to or to , then a norm on is a map , typically denoted by , satisfying the following four axioms: Non-negativity: for every ,. Positive definiteness: for every , if and only if is the zero vector.
Linear algebraLinear algebra is the branch of mathematics concerning linear equations such as: linear maps such as: and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
Dimension (vector space)In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say is if the dimension of is finite, and if its dimension is infinite.
Slide projectorA slide projector is an opto-mechanical device for showing photographic slides. 35 mm slide projectors, direct descendants of the larger-format magic lantern, first came into widespread use during the 1950s as a form of occasional home entertainment; family members and friends would gather to view slide shows. Reversal film was much in use, and supplied slides snapped during vacations and at family events. Slide projectors were also widely used in educational and other institutional settings.
Graded vector spaceIn mathematics, a graded vector space is a vector space that has the extra structure of a grading or gradation, which is a decomposition of the vector space into a direct sum of vector subspaces, generally indexed by the integers. For "pure" vector spaces, the concept has been introduced in homological algebra, and it is widely used for graded algebras, which are graded vector spaces with additional structures. Let be the set of non-negative integers.
Ordered vector spaceIn mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations. Given a vector space over the real numbers and a preorder on the set the pair is called a preordered vector space and we say that the preorder is compatible with the vector space structure of and call a vector preorder on if for all and with the following two axioms are satisfied implies implies If is a partial order compatible with the vector space structure of then is called an ordered vector space and is called a vector partial order on The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping is an isomorphism to the dual order structure.
MathematicsMathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them.
Proof theoryProof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
EndomorphismIn mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space V is a linear map f: V → V, and an endomorphism of a group G is a group homomorphism f: G → G. In general, we can talk about endomorphisms in any . In the , endomorphisms are functions from a set S to itself. In any category, the composition of any two endomorphisms of X is again an endomorphism of X.
