I am self studying linear transformation but i am surprised to be introduced with the ideas of KerT and ImT just after the introduction of the definitions and examples of linear transformation.So my specific question is that what is the motivation of studying KerT and ImT i.e.how do i know beforehand that this abstract collections are going to be helpful enough for the subsequent analysis of spaces and transformations?
My second question is that I have proven two theorems separately
1.Let,V and W be two finite dimensional vector spaces of same dimension over a field F and T:V->W be a linear transformation then T is injective iff T is surjective(proved using rank nullity theorem)
2.Two finite dimensional vector spaces V and W over a field F are isomorphic iff dimV=dimW.
But,is not it so that if i prove the second theorem then dimV=dimW implies T is always bijective and there is no point to prove T is injective iff T is surjective (i.e the first theorem separately)
Any kind of help from any end is welcome