VECTOR SPACES
Vector spaces [ edit ] Main article: Vector space Until the 19th century, linear algebra was introduced through systems of linear equations and matrices . In modern mathematics, the presentation through vector spaces is generally preferred, since it is more synthetic , more general (not limited to the finite-dimensional case), and conceptually simpler, although more abstract. A vector space over a field F (often the field of the real numbers ) is a set V equipped with two binary operations satisfying the following axioms . Elements of V are called vectors , and elements of F are called scalars . The first operation, vector addition , takes any two vectors v and w and outputs a third vector v + w . The second operation, scalar multiplication , takes any scalar a and ...