1
Exact Sequences
Exact sequences of modules, and short/long exact sequences
2
Projective, Injective, and Flat Modules
Definitions and equivalent conditions for projective, injective, and flat modules
3
Bases
Definition of free modules, bases, and the universal property
4
Dual Spaces
Hom of modules, dual modules, and bidual maps
5
Hom and the Tensor Product
Adjunction and exactness of the Hom functor and the tensor product
6
Matrices
Definition and multiplication of matrices over free modules over a general ring
7
Matrices and Linear Maps
Matrix representations of linear maps between free modules and coordinate systems
8
Change of Basis
Square matrices and invertible matrices, transformation of matrices under change of basis
9
Tensor Product of Matrices
The Kronecker product as matrix realization of tensor products
10
Tensor Algebra
Tensor algebra, symmetric algebra, exterior algebra
11
Determinants
The determinant of an endomorphism of a free module and its basic properties
12
Norms and Traces
The norm and trace defined by elements of an algebra
13
Symmetric Tensors
Symmetric group actions, symmetric tensors, and symmetric powers
14
Derivations
Differential modules
15
Differential Modules
Differential modules with derivations on graded algebras