 Sibling to groups, rings, and fields is the class of vector spaces over any given field, constituting the universes of linear algebra. Vector spaces lend themselves to two opposite approaches: axiomatic or abstract, and synthetic or concrete. The axiomatic approach takes fields (whence rings, whence groups) as a prerequisite; it first defines a notion of \(R\)-module as an abelian group with a scalar multiplication over a given ring \(R\), and then defines a vector space to be an \(R\)-module for which \(R\) is a field. The synthetic approach proceeds via the familiar representation of vector spaces over the reals as \(n\)-tuples of reals, and of linear transformations from \(m\)-dimensional to \(n\)-dimensional vector spaces as \(m\times n\) matrices of reals. For the full generality of vector spaces including those of infinite dimension, \(n\) need not be limited to finite numbers but can be any cardinal.