Abstract Vector Spaces

While coordinate vector spaces $F^n$ are the most concrete examples, the concept of a vector space extends to many other mathematical objects: functions, polynomials, sequences, matrices, and solutions to differential equations.

Key idea: If a set satisfies the vector space axioms (closure under addition and scalar multiplication, associativity, commutativity, identity elements, distributivity), then all theorems of linear algebra apply — regardless of what the "vectors" actually are.

This section covers:

• Formal definition and terminology
• Basic properties and consequences of the axioms
• Isomorphism — when two vector spaces are "the same" structurally
• Linear transformation algebra — operations on linear maps
• Eigenvalue theory for linear transformations (independent of matrix representation)