Hint : consider the vector space of sequences of real numbers with the map . Show that has a left inverse which is not a right inverse.
There is a 3 part question that I have been working on.
The first 2 parts are show that for a finite dimensional vector space:
is invertible if and only if and are invertible.
if and only if
Theese two I have successfully proven. It is the last part I am having trouble with which is:
Give an example for each that shows that the statements are false for infinite dimensional vector spaces.
Any help would be appreciated, Thanks!
Thanks for the hint!
Here is what I got
Let
and
will give us the original sequence however
wont. This proves the second
" if and only if " does not hold for infinite dimensional matrices.
However Im still not sure how to prove the first part " is invertible if and only if and are invertible." does not hold for infinite dimensional matrices.
Well you have to be careful; is the identity map, not . You are right that it proves the second part of the problem.
For the first part, notice that is invertible (it's the identity!) but that is not. So the part of the statement which says "If is invertible, then both and are invertible" is false. The part which says "If both and are invertible, then is invertible" is always true; the inverse of is then given by .