Linear algebra

Vector spaces

• Introduction: Vector space 
• Vector space 
• Vector space: properties 
• Proofs for vector spaces 
• Field as a vector space 
• Coordinate spaces 
• Sequence spaces 
• Function spaces 
• Subspace 
• Union and intersection of vector spaces 
• Sum of subspaces 
• Inner direct sum 
• Complements of vector spaces 
• Cosets of a subspace 
• Quotient space 

Linear combinations, generators and bases

• Linear combinations 
• Span 
• Generators 
• Linear independence 
• Basis 
• Steinitz's theorem 
• Dimension 

Linear maps

• Linear map 
• Properties of linear maps 
• Linear continuation 
• Proofs for linear maps 
• Monomorphisms 
• Epimorphisms 
• Isomorphisms 
• Endomorphism and Automorphism 
• Image of a linear map 
• Kernel of a linear map 
• Vector space of a linear map 
• Dual space 
• Exercises 

Matrices

• Introduction: Matrices 
• Matrix of a linear map 
• Definition of a matrix 
• Linear systems and matrices 
• Vector space structure on matrices 
• Matrix multiplication 
• Basis change via matrices 
• Exercises