Matematika Pertemuan 26 Matakuliah: D0024/Matematika Industri II Tahun : 2008

Bina Nusantara A vector space is a set that is closed under finite vector addition and scalar multiplication. The basic example is - dimensional Euclidean space, where every element is represented by a list of real numbers, scalars are real numbers, addition is componentwise, and scalar multiplication is multiplication on each term separately.vector additionscalar multiplicationEuclidean space For a general vector space, the scalars are members of a field, in which case is called a vector space over. field Euclidean -space is called a real vector space, and is called a complex vector space.real vector spacecomplex vector space Vector Space

Bina Nusantara In order for to be a vector space, the following conditions must hold for all elements and any scalars :scalars 1. Commutativity:Commutativity (1) (1) 2. Associativity of vector addition:Associativityvector addition (2) (2) 3. Additive identity: For all, (3) (3) 4. Existence of additive inverse: For any, there exists a such that 5. Associativity of scalar multiplication:Associativity (5) (5) 6. Distributivity of scalar sums:Distributivity 7. Distributivity of vector sums:Distributivity (7) (7) 8. Scalar multiplication identity:

Bina Nusantara A basis of a vector space is defined as a subset of vectors in that are linearly independent and vector space span. Consequently, if is a list of vectors in, then these vectors form a basis if and only if every vector can be uniquely written asvector spacelinearly independentvector space span Vector Space Basis where,..., are elements of the base field. (Outside of pure mathematics, the base field is almost always or, but fields of positive characteristic are often considered in algebra, number theory, and algebraic geometry). A vector space will have many different bases, but there are always the same number of basis vectors in each of them. The number of basis vectors in is called the dimension of. Every spanning list in a vector space can be reduced to a basis of the vector space.algebraic geometryvector space dimension

Bina Nusantara The simplest example of a basis is the standard basis in consisting of the coordinate axes. For example, in, the standard basis consists of two vectors and. Any vector can be written uniquely as the linear combination. Indeed, a vector is defined by its coordinates. The vectors and are also a basis for because any vector can be uniquely written as. The above figure shows, which are linear combinations of the basis. vectorsvector linear combinationvectors vector

Bina Nusantara When a vector space is infinite dimensional, then a basis exists, as long as one assumes the axiom of choice. A subset of the basis which is linearly independent and whose span is dense is called a complete set, and is similar to a basis. When is a Hilbert space, a complete set is called a Hilbert basis.vector spaceaxiom of choicedenseHilbert spaceHilbert basis

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