Hello,

Let a -vector space and .

Prove that invertible invertible.

:)

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- May 5th 2009, 09:52 AMInfophileEquivalence
Hello,

Let a -vector space and .

Prove that invertible invertible.

:) - May 5th 2009, 10:43 AMNonCommAlg
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. - May 6th 2009, 02:46 AMNonCommAlg
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! (Surprised)) 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! (Nod)