linear algebra Introduction

Question 1

field

Answer 1

a set V with binary operations +, * containing 0 and 1 where

(1) 0!=1
(2) (a+b)+c = a + (b+c) (associativity)
(3) a+b=b+a (commutativity)
(4) a+0=a (identity)
(5) there exists and x s.t. a+x=0 (inverse)
(6) (ab)c= a(bc) (associativity)
(7) ab = ba (commutativity)
(8) a1=a (identity)
(9) there exists an x s.t. ax=1 (inverse)
(10) a(b+c)=(ab)+(a*c) (distributivity)

Question 2

degree of a polynomial p(x)

Answer 2

the index n of the largest non-zero coefficient

Question 3

division theorem for polynomials

Answer 3

p(x) divided by q(x) gives

p(x) = s(x)q(x)+r(x)

Question 4

factor of a polynomial p(x)

Answer 4

a polynomial q(x) of degree at least one s.t. there is a third polynomial such that p(x) = q(x)r(x)

Question 5

The remainder theorem

Answer 5

is p(x) is a polynomial and p(a)=0 then (x-a) is a factor of p(x)

Question 6

Bezouts lemma for polynomials (theorem 1.3)

Answer 6

is p(x) and q(x) are polynomials with no common factor then s(x)p(x) + t(x)q(x) = 1