Module 3: Linear Algebra & Matrix Transformations

Welcome to Module 3: Linear Algebra & Matrix Transformations. In this module, we start with simple lists of numbers and graph-paper arrows, build up to dot product directional alignments, discover matrices as coordinate grid transformers, and understand determinants and matrix inverses.

Concepts in this Module

• Concept 01: Vectors, Displacements & Scalar Scaling
  • The Everyday Problem: How does an autonomous robot combine driving forward 2 meters with strafing right 3 meters?
  • Code & Math: Vectors as lists/arrows, vector addition (head-to-tail), scalar multiplication, and basis steps.
• Concept 02: Dot Products, Projections & Alignment
  • The Everyday Problem: How much of our robot’s velocity is pushing directly along the desired path vs. drifting off-track?
  • Code & Math: Dot product u · v, vector projection, perpendicular orthogonality check (dot = 0).
• Concept 03: Matrices as Coordinate Transformers
  • The Everyday Problem: How do we convert driver joystick commands into wheel velocities when the robot is spun at an angle?
  • Code & Math: 2D rotation matrices, matrix-vector multiplication as tracking where unit steps land.
• Concept 04: Determinants, Inverses & Singularity
  • The Everyday Problem: Why does a drive kinematics solver crash with Division by Zero when a robot loses traction or steering locks?
  • Code & Math: Determinant as area scaling, singular matrices (det = 0), and matrix inverses.