Linear Algebra

TopicFieldsComputations involving only integers or only rational numbers are much easierthan those with real numbers. Could other algebraic structures, such as theintegers or the rationals, work in the place of R in the definition of a vectorspace?Yes and no. If we take “work” to mean that the results of this chapter remaintrue then an analysis of the properties of the reals that we have used in thischapter gives a list of conditions that a structure needs in order to “work” in theplace of R.4.1 Definition A field is a set F with two operations ‘+’ and ‘·’ such that(1) for any a, b ∈ F the result of a + b is in F and• a + b = b + a• if c ∈ F then a + (b + c) = (a + b) + c(2) for any a, b ∈ F the result of a · b is in F and• a · b = b · a• if c ∈ F then a · (b · c) = (a · b) · c(3) if a, b, c ∈ F then a · (b + c) = a · b + a · c(4) there is an element 0 ∈ F such that• if a ∈ F then a + 0 = a• for each a ∈ F there is an element −a ∈ F such that (−a) + a = 0(5) there is an element 1 ∈ F such that• if a ∈ F then a · 1 = a• for each element a ≠ 0 of F there is an element a −1 ∈ F such thata −1 · a = 1.The algebraic structure consisting of the set of real numbers along with its