together with the operations of ordinary matrix addition and multiplication of a matrix by a number, forms a vector space over the real numbers.Until now the term "vector" has exclusively referred to "Euclidean vector", examples are position , velocity , etc. The term "vector" applies much more broadly than Euclidean vectors, row or column vectors, etc., see linear algebra and vector space for details. The generators of a Lie group also form a vector space over a field of numbers (e.g. real numbers, complex numbers), since a linear combination of the generators is also a generator. They just live in a different space to the position vectors in ordinary 3d space. The generators form a basis set of V, and the components of the axis-angle and rapidity vectors, , are the coordinates of a Lorentz generator with respect to this basis.In ordinary 3d position space, the position vector is expressed as a linear combination of the Cartesian unit vectors which form a basis, and the Cartesian coordinates are coordinates with respect to this basis.
