Linear Independence - Projective Space of Linear Dependences

Projective Space of Linear Dependences

A linear dependence among vectors v₁, ..., v_n is a tuple (a₁, ..., a_n) with n scalar components, not all zero, such that

If such a linear dependence exists, then the n vectors are linearly dependent. It makes sense to identify two linear dependences if one arises as a non-zero multiple of the other, because in this case the two describe the same linear relationship among the vectors. Under this identification, the set of all linear dependences among v₁, ...., v_n is a projective space.