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 .
Basis (universal algebra)In universal algebra, a basis is a structure inside of some (universal) algebras, which are called free algebras. It generates all algebra elements from its own elements by the algebra operations in an independent manner. It also represents the endomorphisms of an algebra by certain indexings of algebra elements, which can correspond to the usual matrices when the free algebra is a vector space. A basis (or reference frame) of a (universal) algebra is a function that takes some algebra elements as values and satisfies either one of the following two equivalent conditions.
Basis (linear algebra)In mathematics, a set B of vectors in a vector space V is called a basis (: bases) if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B. The elements of a basis are called . Equivalently, a set B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B.
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.
DD tankDD or Duplex Drive tanks, nicknamed "Donald Duck tanks", were a type of amphibious swimming tank developed by the British during the Second World War. The phrase is mostly used for the Duplex Drive variant of the M4 Sherman medium tank, that was used by the Western Allies during and after the Normandy Landings in June 1944. DD tanks worked by erecting a canvas 'flotation screen' around the tank, which enabled it to float in water. DD tanks would use propellers to move forward through water, then lower the flotation screens once they had reached land and fight as an ordinary tank.
Additive categoryIn mathematics, specifically in , an additive category is a C admitting all finitary biproducts. There are two equivalent definitions of an additive category: One as a category equipped with additional structure, and another as a category equipped with no extra structure but whose objects and morphisms satisfy certain equations. A category C is preadditive if all its hom-sets are abelian groups and composition of morphisms is bilinear; in other words, C is over the of abelian groups.
Preadditive categoryIn mathematics, specifically in , a preadditive category is another name for an Ab-category, i.e., a that is over the , Ab. That is, an Ab-category C is a such that every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas: and where + is the group operation. Some authors have used the term additive category for preadditive categories, but here we follow the current trend of reserving this term for certain special preadditive categories (see below).
CoproductIn , the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic to the , which means the definition is the same as the product but with all arrows reversed.
Row and column spacesIn linear algebra, the column space (also called the range or ) 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 or range of the corresponding matrix transformation. Let be a field. The column space of an m × n matrix with components from is a linear subspace of the m-space . 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 is also possible.
ColumnA column or pillar in architecture and structural engineering is a structural element that transmits, through compression, the weight of the structure above to other structural elements below. In other words, a column is a compression member. The term column applies especially to a large round support (the shaft of the column) with a capital and a base or pedestal, which is made of stone, or appearing to be so. A small wooden or metal support is typically called a post.
Mid-ocean ridgeA mid-ocean ridge (MOR) is a seafloor mountain system formed by plate tectonics. It typically has a depth of about and rises about above the deepest portion of an ocean basin. This feature is where seafloor spreading takes place along a divergent plate boundary. The rate of seafloor spreading determines the morphology of the crest of the mid-ocean ridge and its width in an ocean basin. The production of new seafloor and oceanic lithosphere results from mantle upwelling in response to plate separation.
