Linear combinationIn mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.
Product (mathematics)In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors. For example, 21 is the product of 3 and 7 (the result of multiplication), and is the product of and (indicating that the two factors should be multiplied together). When one factor is an integer, the product is called a multiple. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication.
Alternative algebraIn abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have for all x and y in the algebra. Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions. Alternative algebras are so named because they are the algebras for which the associator is alternating. The associator is a trilinear map given by By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal.
Hermann GrassmannHermann Günther Grassmann (Graßmann, ˈhɛʁman ˈɡʏntɐ ˈɡʁasman; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties. His work preceded and exceeded the concept which is now known as a vector space. He introduced the Grassmannian, the space which parameterizes all k-dimensional linear subspaces of an n-dimensional vector space V.
OctonionIn mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface O or blackboard bold . Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.
SedenionIn abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers, usually represented by the capital letter S, boldface S or blackboard bold . They are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic to a subalgebra of the sedenions. Unlike the octonions, the sedenions are not an alternative algebra. Applying the Cayley–Dickson construction to the sedenions yields a 32-dimensional algebra, sometimes called the 32-ions or trigintaduonions.
Dual numberIn algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form a + bε, where a and b are real numbers, and ε is a symbol taken to satisfy with . Dual numbers can be added component-wise, and multiplied by the formula which follows from the property ε^2 = 0 and the fact that multiplication is a bilinear operation. The dual numbers form a commutative algebra of dimension two over the reals, and also an Artinian local ring.
Hilbert's NullstellensatzIn mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert, who proved the Nullstellensatz in his second major paper on invariant theory in 1893 (following his seminal 1890 paper in which he proved Hilbert's basis theorem).
Universal propertyIn mathematics, more specifically in , a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties.
C*-algebraIn mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties: A is a topologically closed set in the norm topology of operators. A is closed under the operation of taking adjoints of operators.
