Matrix Algebra and Invertibility
Composition, Products, and LU Decomposition
Master matrix multiplication as composition of transformations, understand invertibility through 10 equivalent conditions, and learn the LU decomposition algorithm.
Sections
Composition of Linear Maps
10 min · 1 quiz question · Interactive diagram
Defining Matrix-Matrix Multiplication
10 min · 1 quiz question
The Column Perspective on Matrix Products
10 min · 1 quiz question · Interactive diagram
The Entry Perspective: Dot Products
9 min · 1 quiz question
Algebraic Properties of Matrix Multiplication
9 min · 1 quiz question
Caveats: Non-Commutativity and Zero Products
8 min · 1 quiz question
Rank of a Product
9 min · 1 quiz question
Right Invertibility and Surjectivity
9 min · 1 quiz question
Left Invertibility and Injectivity
9 min · 1 quiz question
The Invertible Matrix Theorem
12 min · 1 quiz question · Interactive diagram
Non-Singular Matrices and Unique Inverses
9 min · 1 quiz question
Permutation and Triangular Matrices
10 min · 1 quiz question
LU Decomposition and Efficient Solving
14 min · 2 quiz questions · Interactive diagram