Thread: equivalence relation of ordered basis

Let B1={v1,...,vn) and B2={u1,...,un} be two ordered bases for R^n..Then B1 is related to B2 if det(A)>0. A is the associated n(cross)n matrix..

Let B={v1,...,vn} be ordered basis of R^n..Let *:{1,...,n}-->{1,...,n} be a permutation..Show that the ordered basis B* = {v*(1),...,v*(n)} belongs to the equivalence class of B if and only of * is an even permutaion..

Please help!

Swapping two rows or columns in a determinant multiplies the determinant by -1.