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. 🌐vector space diagramcomputer graphics vectormachine learning vectors