Vector Spaces & Subspaces

Linear Algebra for Machine Learning · Axiom Academy

A formal mathematical structure A vector space is a set of vectors closed under addition and scalar multiplication, with certain properties. A vector space V over a field F (usually ℝ) is a set with two operations (addition and scalar multiplication) satisfying 10 axioms. Closure under addition: If u, v are in V, then u + v is in V. Closure under scalar mult: If v is in V and c is a scalar, then cv is in V. Zero vector: There exists a zero vector 0 such that v + 0 = v for all v. Additive inverses: For every v, there exists -v such that v + (-v) = 0. Commutativity & Associativity: Addition is commutative and associative. Example: ℝⁿ is a vector space. Any two n-dimensional vectors sum to an n-dimensional vector. Any scalar multiple of an n-dimensional vector is n-dimensional. Vector spaces provide a unified framework where we can apply linear algebra tools consistently. If we have a vector space, we know operations are well-defined. ℝⁿ: n-dimensional Euclidean space Polynomial spaces: polynomials of degree ≤ n Function spaces: continuous functions on an interval The beauty is: all these spaces follow the same rules, so we can use the same mathematical techniques!

This is the written version of the interactive lesson above. See the full Linear Algebra for Machine Learning course.