Springer UTM, 1996

Linear Algebra

"You are probably about to begin your second exposure to linear algebra. Unlike your first brush with the subject, which probably emphasized Euclidean spaces and matrices, we will focus on abstract vector spaces and linear maps. These terms will be defined later, so don't worry if you don't know what they mean. This books starts from the beginning of the subject, assuming no knowledge of linear algebra. The key point is that you are about to immerse yourself in serious mathematics, with an emphasis on you attaining a deep understanding of the definitions, theorems, and proofs."

Chapter 1 : Vector SpacesEdit

Chapter 2 : Finite-Dimensional Vector SpacesEdit

Chapter 3 : Linear MapsEdit

Chapter 4 : PolynomialsEdit

Chapter 5 : Eigenvalues and EigenvectorsEdit

Invariant SubspacesEdit

Polynomials Applied to OperatorsEdit

Upper-Triangular MatricesEdit

  • "W invariant under T" means that $ TW \subseteq W $.

Diagonal MatricesEdit

Invariant Subspaces on Real Vector SpacesEdit


Chapter 6 : Inner-Product SpacesEdit

Chapter 7 : Operators on Inner-Product SpacesEdit

Chapter 8 : Operators on Complex Vector SpacesEdit

Chapter 9 : Operations on Real Vector SpacesEdit

Chapter 10 : Trace and DeterminantEdit