Linear Maps & Matrices
A 20-module path through linear maps & matrices, in study order from foundations to the frontier.
Modules
- Linear Maps: Definition and Basic Properties
- Examples of Linear Maps
- Linear Maps Determined by Action on a Basis
- The Null Space and Injectivity
- The Range and Surjectivity
- The Rank-Nullity Theorem
- Consequences of Rank-Nullity for Finite Dimensions
- The Matrix of a Linear Map
- Coordinate Vectors and Matrix Action
- The Isomorphism L(V,W) ≅ Matrices
- Matrix Multiplication as Composition
- Invertibility of Linear Maps
- Isomorphisms and Isomorphic Vector Spaces
- Change of Basis and the Change-of-Basis Matrix
- Similar Matrices
- Invertible Matrices
- The Dual Space and Dual Bases
- Dual Maps and the Transpose
- The Double Dual and Natural Isomorphism
- Elementary Matrices and Row Operations