Dashboard

Now you are in the subtree of Container for Linear Algebra project. 

Vector spaces

Created over 8 years ago, updated 10 days ago

Vector spaces are one of the foundational structures in linear algebra. A vector space is a set of objects called vectors that can be added together and multiplied by scalars (numbers from a field, typically (\mathbb{R}) or (\mathbb{C})), satisfying a specific set of axioms.

Overview

Vector spaces generalize the familiar notion of Euclidean space (\mathbb{R}^n) to more abstract settings. The key insight is that many mathematical objects --- tuples of numbers, matrices, polynomials, functions, sequences --- can all be treated as vectors once we define appropriate addition and scalar multiplication operations.

Key Concepts in This Section

  • Coordinate vector spaces ((\mathbb{R}^n), (\mathbb{C}^n)): The prototypical examples where vectors are ordered tuples of numbers.
  • Axioms of a vector space: The ten properties that any vector space must satisfy.
  • Linear combinations, spans, and subspaces: Tools for building and analyzing subsets of vector spaces.
  • Linear independence and bases: Understanding minimal generating sets and the notion of dimension.
  • Linear transformations: Structure-preserving maps between vector spaces.
  • Orthogonality and projection: Geometric concepts that extend to inner product spaces.
  • Abstract vector spaces: Examples beyond coordinate spaces, including function spaces, polynomial spaces, and matrix spaces.

Vector spaces provide the language for discussing systems of linear equations, linear transformations, eigenvalues, and much of modern mathematics and its applications in physics, engineering, and computer science.