Let be vectors in ; let be a matrix . If is an arbitrary -tuple from the set , show that
,
where
for
Please help or give a hint
write out the definition of the wedge product. then write out your vectors in the basis ej. Then using what you have above, and the fact that each phi i is linear, you realise that the only non zero component in your sum is when each phi i is applied to just ei. ie. phi1(e1).phi2(e2)...but this is just the definition of the determinant of the matrix X