In the previous chapters, we established what vectors and matrices are and how to perform operations on them. Now, we will apply these tools to solve one of the most frequent problems in computation: systems of linear equations. Many tasks in machine learning, such as fitting a model to data, can be formulated as a system of equations that must be solved.

This chapter shows how to represent a system of multiple equations with multiple unknowns in the compact matrix form:

$$Ax = b$$

Here, $A$ is a matrix of known coefficients, $x$ is a vector of the unknown variables we want to find, and $b$ is a vector of known outcomes. Our objective is to find the solution vector $x$.

To do this, you will learn about:

• The matrix inverse and its function in isolating the unknown vector $x$.
• How to calculate the determinant of a matrix and use it to check if a unique solution exists.
• The difference between invertible (non-singular) and non-invertible (singular) matrices.
• Methods for solving the equation $Ax = b$ using NumPy's linear algebra library.