Introduction: what vector spaces are and why they’re fundamental in mathematics. Useful as a foundation for data science, geometry, algebra, and many more topics.
A vector space over a field 𝔽 is a set V together with two operations satisfying certain axioms:
- Vector addition: for u, v ∈ V, u + v ∈ V
- Scalar multiplication: for α ∈ 𝔽 and v ∈ V, α v ∈ V
- Axioms: associative, commutative, existence of 0, inverses, distributivity, identity, etc.
In symbols:
\[ \forall u, v, w \in V,\; (u + v) + w = u + (v + w), \quad u + v = v + u, \quad \dots \]
Common examples:
- ℝⁿ with standard addition and scalar multiplication
- Polynomials of degree ≤ d, with coefficient‑wise operations
- Continuous functions on [a,b], or other function spaces
A subspace W ⊆ V must be closed under addition and scalar multiplication, contain 0, etc. A basis is a linearly independent spanning set. The number of basis vectors is the dimension of V.
Example in ℝ³: basis { (1,0,0), (0,1,0), (0,0,1) } — dimension = 3.