Phase 0: Linear Algebra Foundations · ~120 minutes · Python

Linear Systems

Solving Ax = b is the oldest problem in mathematics that still runs your neural network.

What you will learn

Solve Ax = b using Gaussian elimination with partial pivoting and back substitution
Factor matrices with LU, QR, and Cholesky decompositions and explain when each is appropriate
Derive the normal equations for least squares and connect them to linear and ridge regression
Diagnose ill-conditioned systems using the condition number and apply regularization to stabilize them

Introduction

Type: Build
Language: Python
Prerequisites: Linear Algebra Intuition, Matrix Transformations
Time: ~120 minutes

Objective

Learning objectives

Solve Ax = b using Gaussian elimination with partial pivoting and back substitution
Factor matrices with LU, QR, and Cholesky decompositions and explain when each is appropriate
Derive the normal equations for least squares and connect them to linear and ridge regression
Diagnose ill-conditioned systems using the condition number and apply regularization to stabilize them

Unlock the full lesson

You've read the first 2 sections. The rest of this lesson covers The Problem, The Concept, Build, Check Yourself, Key Terms & Next — plus a hands-on lab, quiz, and project artifact.

Create a free account — the opening phases of 29 of 35 courses are free, no credit card.