Determinants, An Introduction

Introduction

Every matrix implements a linear map between vector spaces. The determinant of the matrix measures the change in area or volume as the domain is mapped onto the range. Does a square foot become a square inch, or a square mile? Or is it squashed down to a line or a point?

A determinant of 0 compresses n dimensions into a smaller subspace, and the function is not 1-1; it cannot be reversed. The determinant is 0 iff the matrix is singular. We'll prove all this as we go.

Here's a simple example in one dimension. Consider the linear function f(x) = kx, with matrix k. The determinant of this matrix is k, and sure enough, an inch in the domain becomes k inches in the range. If k is negative the space is reflected. If k is 0 the entire line is squashed down to the origin, and the function cannot be reversed.