A STUDY ON VECTOR SPACE WITH ADDITIONAL STRUCTURE

Author (s) : V.Nandhini, & V.Rajeshwari, Dr.P,Soumiya

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Abstract :
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. A vector space V is a set with a linear structure. This means we can add elements of the vector space or multiply elements by scalars (real numbers) to obtain another element. A familiar example of a vector space is Rⁿ. Given x = (x₁,..., x_n) and y = (y₁,...,y_n) in Rⁿ, we can form a new vector  x + y = (x₁ + y₁,..., x_n + y_n) ∈ Rⁿ. Similarly, given r ∈ R, we can form rx = (rx₁,..., rx_n) ∈ Rⁿ . A more general kind of abstract vector space is obtained if one admits that the basis has infinitely many elements. In this case, the vector space is called infinite-dimensional and its elements are the formal expressions in which all but a finite number of coefficients are equal to zero.

Keywords : Vector space, Structure, Real vector space, Complex vector space

Conference Date : 30/08/2019

Pages : 331

Paper ID : 19322