A vector subspace of R^n is a subset that is closed under linear combinations, meaning it must contain the zero vector and any scalar multiple or sum of its elements must remain within the subset. This concept relates to matrix theory through canonical examples such as the null space (vectors mapped to zero by a matrix) and column space (span of a matrix's columns), and determining whether an arbitrary subset qualifies as a subspace requires verifying closure under linear combination — often by rewriting a defining condition as a homogeneous linear (matrix) equation, checking closure under scalar multiplication, or testing for the necessary presence of the zero vector.
A vector subspace of R^n is a subset that is closed under linear combinations, meaning it must contain the zero vector and any scalar multiple or sum of its elements must remain within the subset. Th…