Fill content

# Abstract vector spaces

Put content here.# 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)

# Parents

* Vector spaces