Thread: Equivalence

Hello,

Let a -vector space and .

Prove that invertible invertible.

Originally Posted by Infophile

Hello,

Let a -vector space and .

Prove that invertible invertible.

because of the symmetry, we only need to prove let's show by

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!