701 - CHAPTER 7 VECTOR SPACES There are many mathematical...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
1 CHAPTER 7 VECTOR SPACES There are many mathematical systems in which the notions of addition and multiplication by scalars are defined. Example: R n = {all n -vectors with real entries}. Example: R m × n = {all m × n matrices with real entries}. Example: P n = {all polynomials in the variable x with degree n }. Example: P = {all polynomials in the variable x }. Example: F ( R ) = {all functions mapping from R to R }. All these mathematical systems share some common fundamental properties ( axioms ), which may be used to derive a framework ( general theory ) of new concepts ( definitions ) and useful conclusions ( theorems ) that apply to each of these mathematical systems. Definition. A field F is a set in which two operations “+” (addition) and “·” (multiplication) are defined so that for any a , b F , there are unique elements a + b (sum of a and b ) and a · b (product of a and b ) in F , and for all a , b , c F , the following conditions hold: 1. a + b = b + a and a · b = b · a (commutativity). 2. ( a + b ) + c = a + ( b + c ) and ( a · b c = a ·( b · c ) (associativity). 3. distinct elements 0 and 1 in F such that 0 + a = a and 1· b = b (existence of identity elements). 4.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 6

701 - CHAPTER 7 VECTOR SPACES There are many mathematical...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online