Handbook of Linear Algebra, 2nd Edition

Chapter 65

Linear Algebra and Mathematical Physics

Lorenzo Sadun

University of Texas

65.1 Introduction

Linear algebra appears throughout physics. Linear differential equations, both ordinary and partial, appear through classical and quantum physics. Even where the equations are nonlinear, linear approximations are extremely powerful.

Two big ideas underpin linear analysis in physics. The first is the Superposition Principle. Suppose we have a linear problem where we need to compute the output for an arbitrary input. If there is a solution to the problem with input I1 and output O1, a solution with input I2 and output O2, etc., then the response to the input $c_1{I}_1+\cdots +c_k{I}_k$ is $c_1{O}_1+\cdots c_k{O}_k$. It is, therefore, enough to solve our problem for a limited ...