Some Probability Problems In Linear Algebra

The study on primes and coprime integers goes back to50years ago. In1969, Dirichlet discovered that the probability of two random chosen integers being coprime, then this result was generalized as ζ-1(2)) then this result was generalized as:the probability of n integers being coprime is ζ-1(n), where C(n) is the famous Reimann’s Zeta function. In this paper, we commute the probability that n polynomials over a finite field are k-wise relatively co-prime, and as a special cake for k=2, we obtain the probability that n such polynomials are mutually co-prime.It is well known that n integers are co-prime if and only if they can form a row or column of an invertible integral matrix, which indicates the natural relation between co-prime integers and invertible integer matrices. There are many works on this topics. For example, Maze, Rosenthal and Wagner obtained that the probability that a k x n integral matrix can be extended to an invertible n x n matrix over the integers is Πjn=n-k+1ζ(j)-1. In this dissertation, we generalize their result to a more general case. More precisely, we study the probability that an r x s integer matrices can be extended to an invertible integer matrices, then we consider the similar problem for the polynomial ring over a finite field.