Linear Algebra: A Modern Introduction

Description

Linear algebra is a field of mathematics that is universally agreed to be a prerequisite to a deeper understanding of machine learning. Although linear algebra is a large field with many esoteric theories and findings, the nuts and bolts tools and notations taken from the field are practical for machine learning practitioners. With a solid foundation of what linear algebra is, it is possible to focus on just the good or relevant parts. In algebra, the determinant is a special number associated to any square matrix, that is to say, a rectangular array of numbers where the (finite) number of rows and columns are equal. The fundamental geometric meaning of a determinant is a scale factor for measure when the matrix is regarded as a linear transformation. Thus a 2 × 2 matrix with determinant 2 when applied to a set of points with finite area will transform those points into a set with twice the area. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra. A matrix is invertible if and only if its determinant is non-zero. Linear Algebra is the heart of applied science but there are divergent views concerning its meaning. The field of Linear Algebra is more beautiful and more fundamental than its rather dull name may suggest. More beautiful because it is full of powerful ideas that are quite unlike those normally emphasized in a linear algebra course in a mathematics department. This textbook on Linear Algebra provides a modern, elementary introduction to linear algebra and some of its interesting applications, suitable for the students of mathematics and allied fields.