1 Jared Kirkham Math 308 Project Autumn 2001

I have chosen to explore how linear algebra can be applied to genetics. More specifically, I will focus entirely on the phenomena of autosomal inheritance. My goal is to show how linear algebra can be used to predict the genotype distribution of a particular trait in a population after any number of generations from only the genotype distribution of the initial population. In order to perform such an analysis, I will use several critical concepts from linear algebra, including the following: Difference equations, diagonalization of a matrix, inverse of a matrix, eigenvalues, and eigenvectors. Explanation of the application Background: Genetics is the study of inheritance, or the transmission of traits from one generation to the next. There are several modes of inheritance; however, this project will focus only on autosomal inheritance. In autosomal inheritance, each heritable trait is assumed to be governed by a single gene on a chromosome. There are typically two different forms, or alleles of a gene (denoted by A and a). Each individual in a population carries a pair of alleles, which may be similar or different. These pairs are called an individual’s genotype, and there are three possible genotypes for a particular trait: AA, Aa, or aa. It is the genotype that determines how the trait controlled by the gene is manifested in the individual. For example, in humans, eye coloration is controlled through autosomal inheritance. Genotypes AA and Aa have brown eyes, and genotype aa has blue eyes. In such a case, the A allele is dominant over the allele, or that the allele is recessive to the A allele, since genotype Aa has the same outward trait as genotype AA.