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## Main Question or Discussion Point

Im not sure whether this is a "Homework Question", but it is a question regarding the proof of the Inverse Function Theorem. It starts like this:

Let k be the linear transformation Df(a). Then k is non-singular, since det(f '(a)) != 0. Now D((k^-1(f(a))) = D(k^-1)(f(a)) (Df(a)) = k^-1 (Df(a)) is the identity linear transformation.

Heres what i dont understand:

If the theorem is true for k^-1 (f) then it is clearly true for f. Therefore we may assume at the outset the k is the identity.

Can anyone explain this?

Let k be the linear transformation Df(a). Then k is non-singular, since det(f '(a)) != 0. Now D((k^-1(f(a))) = D(k^-1)(f(a)) (Df(a)) = k^-1 (Df(a)) is the identity linear transformation.

Heres what i dont understand:

If the theorem is true for k^-1 (f) then it is clearly true for f. Therefore we may assume at the outset the k is the identity.

Can anyone explain this?