relations between vector space, span, subspace, affine set, hyperplane

Characteristic	Vector Space	Span	Subspace	Affine Set	Hyperplane
Definition	A set of vectors that can be added together and multiplied by scalars	The set of all linear combinations of a set of vectors.	A subset of a vector space	A set of vectors that can be obtained by adding a single vector to the span of a set of vectors. can but no need to pass through the origin linear + scalar = affine	A subspace of a vector space, dim hype = dim vec space - 1
Linear algebra attributes	closure	closure	closure contain the zero vector of the vector space.	closure contain min 1 vector not in the span of the vectors being added to. not necessarily preserve vector addition and scalar multiplication	closure contain min 1 vector not in span of vectors defining hyperplane