Chapter 3

Matrices and Linear Maps

Rank, Nullity, and the Structure of Transformations

Understand matrices as linear transformations, master the column space and null space, and discover the Rank-Nullity theorem that ties everything together.

17 sections~152 minutes

Sections

3.1

Matrix Basics

8 min · 1 quiz question

3.2

Matrix-Vector Multiplication

10 min · 1 quiz question · Interactive diagram

3.3

Linear Maps as Functions

9 min · 1 quiz question

3.4

Matrices and Linear Maps: The Equivalence

10 min · 1 quiz question

3.5

Column Span and Rank

10 min · 1 quiz question

3.6

Surjectivity: Hitting Every Output

9 min · 1 quiz question

3.7

The Null Space

10 min · 1 quiz question

3.8

Injectivity: No Collisions

9 min · 1 quiz question

3.9

Injectivity and Nullity

8 min · 1 quiz question

3.10

Injectivity and Linear Independence

8 min · 1 quiz question

3.11

Bijectivity and Square Matrices

9 min · 1 quiz question

3.12

The Collapse Theorem

8 min · 1 quiz question

3.13

Linear Systems as Matrix Equations

8 min · 1 quiz question

3.14

Solvability: When Does Ax = b Have a Solution?

8 min · 1 quiz question

3.15

Homogeneous and Inhomogeneous Systems

8 min · 1 quiz question

3.16

Solution Set Structure

8 min · 1 quiz question

3.17

The Rank-Nullity Theorem

12 min · 2 quiz questions · Interactive diagram

Start Chapter 3