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Linear Hull
In mathematics, the linear span (also called the linear hull or just span) of a set S of elements of a vector space V is the smallest linear subspace of V that contains S. It is the set of all finite linear combinations of the elements of , and the intersection of all linear subspaces that contain S. It is often denoted pp. 29-30, §§ 2.5, 2.8 or \langle S \rangle. For example, in geometry, two linearly independent vectors span a plane. To express that a vector space is a linear span of a subset , one commonly uses one of the following phrases: spans ; is a spanning set of ; is spanned or generated by ; is a generator set or a generating set of . Spans can be generalized to many mathematical structures, in which case, the smallest substructure containing S is generally called the substructure ''generated'' by S. Definition Given a vector space over a field , the span of a set of vectors (not necessarily finite) is defined to be the intersection of all subspac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis For A Plane
Basis is a term used in mathematics, finance, science, and other contexts to refer to foundational concepts, valuation measures, or organizational names; here, it may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items * Basis point, 0.01%, often used in the context of interest rates * Basis swap, an interest rate swap * Cost basis, in income tax law, the original cost of property adjusted for factors such as depreciation * Tax basis, cost of an asset Securities markets and trading strategies * Basis trading, a trading strategy consisting of the purchase of a security and the sale of a similar security Fixed income markets: * Treasury basis trade, a leveraged arbitrage strategy exploiting price differences between Treasury securities and futures contracts * Index arbitrage, a strategy that exploits price differences between a stock index and its futures contract Commodities and physical assets: * C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuously
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. It is sometimes said that a statement is vacuously true because it does not really say anything. For example, the statement "all cell phones in the room are turned off" will be true when no cell phones are present in the room. In this case, the statement "all cell phones in the room are turned ''on''" would also be vacuously true, as would the conjunction of the two: "all cell phones in the room are turned on ''and'' turned off", which would otherwise be incoherent and false. More formally, a relatively well-defined usage refers to a conditional statement (or a universal conditional statement) with a false antecedent. One example of such a statement is "if Tokyo is in Spain, then the Eiffel Tower is in Bolivia". Such statements are considered vacuous truths because ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Module
In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z-module) that is generated by one element. Definition A left ''R''-module ''M'' is called cyclic if ''M'' can be generated by a single element i.e. for some ''x'' in ''M''. Similarly, a right ''R''-module ''N'' is cyclic if for some . Examples * 2Z as a Z-module is a cyclic module. * In fact, every cyclic group is a cyclic Z-module. * Every simple ''R''-module ''M'' is a cyclic module since the submodule generated by any non-zero element ''x'' of ''M'' is necessarily the whole module ''M''. In general, a module is simple if and only if it is nonzero and is generated by each of its nonzero elements. * If the ring ''R'' is considered as a left module over itself, then its cyclic submodules are exactly its left principal ideals as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid
In combinatorics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid Axiomatic system, axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or ''flats''. In the language of partially ordered sets, a finite simple matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terms used in both linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, topology, combinatorial optimization, network theory, and coding theory. Definition There are many Cryptomorphism, equivalent ways to define a (finite) matroid. Independent sets In terms of independence, a finite matroid M is a pair (E, \mathcal), where E is a finite set (called the ''gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family (S_i)_ of nonempty sets (S_i as a nonempty set indexed with i), there exists an indexed set (x_i)_ such that x_i \in S_i for every i \in I. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. The axiom of choice is equivalent to the statement that every partition has a transversal. In many cases, a set created by choosing elements can be made without invoking the axiom of choice, particularly if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available — some distinguishing property that happens to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Independence
In the theory of vector spaces, a set (mathematics), set of vector (mathematics), vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts are central to the definition of Dimension (vector space), dimension. A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space. Definition A sequence of vectors \mathbf_1, \mathbf_2, \dots, \mathbf_k from a vector space is said to be ''linearly dependent'', if there exist Scalar (mathematics), scalars a_1, a_2, \dots, a_k, not all zero, such that :a_1\mathbf_1 + a_2\mathbf_2 + \cdots + a_k\mathbf_k = \mathbf, where \mathbf denotes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problem (mathematics education), word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called a power product or primitive monomial, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, x^2yz^3=xxyzzz is a monomial. The constant 1 is a primitive monomial, being equal to the empty product and to x^0 for any variable x. If only a single variable x is considered, this means that a monomial is either 1 or a power x^n of x, with n a positive integer. If several variables are considered, say, x, y, z, then each can be given an exponent, so that any monomial is of the form x^a y^b z^c with a,b,c non-negative integers (taking note that any exponent 0 makes the corresponding factor equal to 1). # A monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A primitive monomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Dependency
In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts are central to the definition of dimension. A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space. Definition A sequence of vectors \mathbf_1, \mathbf_2, \dots, \mathbf_k from a vector space is said to be ''linearly dependent'', if there exist scalars a_1, a_2, \dots, a_k, not all zero, such that :a_1\mathbf_1 + a_2\mathbf_2 + \cdots + a_k\mathbf_k = \mathbf, where \mathbf denotes the zero vector. This implies that at least one of the scalars is nonzero, say a_1\ne 0, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Basis
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane \mathbb^2 formed by the pairs of real numbers, the standard basis is formed by the vectors \mathbf_x = (1,0),\quad \mathbf_y = (0,1). Similarly, the standard basis for the three-dimensional space \mathbb^3 is formed by vectors \mathbf_x = (1,0,0),\quad \mathbf_y = (0,1,0),\quad \mathbf_z=(0,0,1). Here the vector e''x'' points in the ''x'' direction, the vector e''y'' points in the ''y'' direction, and the vector e''z'' points in the ''z'' direction. There are several common notations for standard-basis vectors, including , , , and . These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors). These vectors are a basis in the sense that any othe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
