Let a -vector space and .
Prove that invertible invertible.
suppose that is invertible. then there exists such that which gives us:
multiplying (1) by from the left gives us therefore because
thus hence: this shows that is a right inverse of
this time multilpy (1) by from the right to get and so because
thus: and hence i.e. is also a left inverse of Q.E.D.
this is true in any ring with unity R, i.e. for any two elements if the element is invertible, then is invertible too and the proof is exactly as i did.
but there's an interesting story behind this. i showed that if is the inverse of then would be the inverse of professor Tsit Yuen Lam in one
of his books mentions that Kaplansky taught him a way to remember this: since is the inverse of we write:
(geometric series! ) then we'll have:
this is anything but a valid solution. it's actually a completely invalid way which gives a correct answer! Kaplansky just wanted to teach his student a little trick!