Having explored vectors, matrices, and their fundamental operations, we now turn to one of their primary applications: representing and solving systems of linear equations. Many problems in science, engineering, and machine learning can be formulated as finding a set of unknowns that satisfy multiple linear constraints simultaneously.

This chapter introduces how to express these systems concisely using matrix notation, often written as $Ax = b$. We will look at what constitutes a solution to such a system. You will learn about the identity matrix and its properties, leading to the concept of the matrix inverse, a tool analogous to division in scalar algebra. We will discuss when an inverse exists and how it can be used, at least conceptually, to find the solution vector $x$. Finally, we will cover how to use Python's NumPy library to calculate matrix inverses and, more importantly, to solve systems of linear equations efficiently using specialized functions.

By the end of this chapter, you will understand:

• How to represent systems of linear equations in the matrix form $Ax = b$.
• The role of the identity matrix and the definition of a matrix inverse.
• The conditions under which a matrix inverse exists.
• How to compute matrix inverses and solve linear systems using NumPy.