Thread: transformation question

Question:

If T is a linear operator on a vector space V(F) such that (zero map), then show that T is invertible.


i know that to show invertible i have to show 1-1 and onto, but i have no idea how to begin showing this ... any help?

. Now see if the kernel has any vectors different from zero.

Slight variation:

Can you commute T and I-T?

Yes, of course: . Any linear operator from a vector space to itself commutes with itself and the identity.

Originally Posted by HallsofIvy

Slight variation:

Originally Posted by ModusPonens

Can you commute T and I-T?

Moreover, judging from the level of the question one would guess that one can assume that the vector space is finite dimensional from where implies is surjective from where (by finite dimensionality) bijectivity follows.