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
Dual Spaces
Hom of modules, dual modules, and the bidual map
4
Hom and the Tensor Product
Adjunction and exactness of the Hom functor and the tensor product
5
Matrices
Definition and multiplication of matrices over free modules over a general ring
6
Matrices and Linear Maps
Matrix representations of linear maps between free modules and coordinate systems
7
Change of Basis
Square matrices and invertible matrices, transformation of matrices under change of basis
8
Tensor Algebras
Tensor algebras, symmetric algebras, exterior algebras
9
Determinants
The determinant of an endomorphism of a free module and its basic properties
10
Norms and Traces
The norm and trace defined by elements of an algebra
11
Differentiation
Differential modules
12
Differential Modules
Differential modules with derivations on graded algebras
13
Symmetric Tensors
The action of the symmetric group, symmetric tensors, and symmetric powers