Vector spaces, also known as linear spaces, are foundational concepts in linear algebra. They consist of a set of vectors and a field of scalars (e.g., real or complex numbers) that satisfy specific axioms, such as closure under addition and scalar multiplication, associativity, and distributivity. 📚

Key Properties

  • Closure: Adding vectors or multiplying by scalars results in another vector in the same space.
  • Commutativity: Vector addition is commutative (u + v = v + u).
  • Identity Elements: Contains a zero vector and multiplicative identity (1·v = v).
  • Inverse Elements: Every vector has an additive inverse (-v).

Applications

Vector spaces are widely used in:

  • Physics (e.g., forces, velocities in mechanics)
  • Computer Graphics (3D modeling and transformations)
  • Machine Learning (feature spaces and optimization)
  • Quantum Mechanics (state vectors in Hilbert spaces)

Further Learning

For deeper insights, explore our guide on Linear Algebra Basics. 🌐


computer graphics vector

machine learning vectors