Rattle space equations

Axiom Statement Associativity of vector addition u + (v + w) = (u + v) + w Commutativity of vector addition u + v = v + u Identity element of vector addition There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V. Inverse elements of vector addition For every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0. Compatibility of scalar multiplication with field multiplication a(bv) = (ab)v [nb 3] Identity element of scalar multiplication 1v = v, where 1 denotes the multiplicative identity in F. Distributivity of scalar multiplication with respect to vector addition   a(u + v) = au + av Distributivity of scalar multiplication with respect to field addition (a + b)v = av + bv Linear independence The elements of a subset G of a F-vector space V are said to be linearly independent if no element of G can be written as a linear combination of the other elements of G. Equivalently, they are linearly independent if two linear combinations of elements of G define the same element of V if and only if they have the same coefficients. Also equivalently, they are linearly independent if a linear combination results in the zero vector if and only if all its coefficients are zero. Functions from any fixed set Ω to a field F also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions f and g is the function (𝑓+𝑔){\displaystyle (f+g)} given by [drop] [outro]