# Linear Systems

In the Calculus for Machine Learning course, we explored the framework of calculus and used it to:

- Understand the slope of linear functions.
- Understand the derivative (slope as a function) of nonlinear functions.
- Find extreme values in nonlinear functions.

While we learned the basics of slope through linear functions, we primarily focused on nonlinear functions in the last course.

In this course, we’ll focus on understanding linear functions. Specifically, we’ll explore the framework of linear algebra, which provides a way to represent and understand the solutions to systems of linear equations. A system of linear equations consists of multiple, related functions with a common set of variables.

In this lesson, we’ll learn how to use matrices to solve systems of linear functions using a number of different techniques. You will learn how to solve linear systems using Gaussian elimination, reducing the system to echelon form and other techniques. You will learn how to represent a function in general form as well as utilizing NumPy to represent an augmented matrix.

As you work through each concept, you’ll get to apply what you’ve learned from within your browser so that there’s no need to use your own machine to do the exercises. The Python environment inside of this course includes answer checking so you can ensure that you’ve fully mastered each concept before learning the next concept.

## Objectives

- Learn how to represent a problem as a linear system.
- Learn how to solve linear systems by elimination.

## Lesson Outline

- Overview Of Linear Algebra
- Solving Linear Systems By Elimination
- Representing Functions In General Form
- Representing An Augmented Matrix In NumPy
- Matrix Representation Of The Solution
- Row Operations
- Simplifying Matrix To Echelon Form
- Row Reduced Echelon Form
- Next Steps
- Takeaways

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