VECTOR SPACES CHAPTER 1 INTRODUCTION A vector

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VECTOR SPACES CHAPTER 1 INTRODUCTION A vector space or a linear space is a group of objects called vectors, added collectively and multiplied (scaled) by numbers, called scalars. Scalars are usually considered to be real numbers. But

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VECTOR SPACES CHAPTER 1<br>
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INTRODUCTION A vector space or a linear space is a group of objects called vectors, added collectively and multiplied (“scaled”) by numbers, called scalars. Scalars are usually considered to be real numbers. But there are few cases of scalar multiplication by rational numbers, complex numbers, etc. with vector spaces. The methods of vector addition and scalar multiplication must satisfy specific requirements such as axioms. Real vector space and complex vector space terms are used to define scalars as real or complex numbers.<br>
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BINARY COMPOSITION Just as we get a number when two numbers are  either added or subtracted or multiplied or are divided. The binary operations associate any two elements of a set. The resultant of the two are in the same set. Binary operations on a set are calculations that combine two elements of the set (called operands) to produce another element of the same set.
The binary operations * on a non-empty set A are functions from A × A to A. The binary operation, *: A × A → A. It is an operation of two elements of the set whose domains and co-domain are in the same set.<br>