Calculus Revisited: Multivariable Calculus

Part IV: Matrix Algebra

This part of the course extends the concept of inverse functions to the case where y = f(x) with y = (y₁, y₂, …, y_n) and x = (x₁, x₂, …, x_n). In more user-friendly terms this part asks us to determine when and how the system of equations that expresses y₁, y₂, … and y_n as functions of x₁, x₂, and x_n can be “inverted” to express x₁, x₂, and x_n as functions of y₁, y₂, … and y_n . This motivates the study of matrix algebra since the process of inverting an n x n square matrix is used to show how we decide whether a function f(x₁, x₂, …, x_n) has an inverse and how we find the inverse function if it exists.

Lecture 1: Linearity Revisited

Lecture 2: The "Game" of Matrices

Lecture 3: Inverting a Matrix

Lecture 4: Inverting More General Systems of Equations

Lecture 5: Maxima and Minima in Several Variables