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


For your second question: no. Suppose $V$ and $W$ are vector spaces of the same dimension. I can construct a linear transformation $T:V \to W$ which is not bijective; for instance, let $T(v)=0_W$ for all $v \in V$. The issue is that if $V$ and $W$ are isomorphic, it doesn't mean that every linear transformation between them is an isomorphism.

For your first question: Didn't you already use the notions of kernel and image in proving the above two theorems? :)

It's impossible to completely communicate their usefulness before you have some more experience. But here's a little bit: Suppose we have a linear map $T:V \to W$. Often there are quite a lot of elements in $V$, and so the full data of what the linear map does is vast. We look for ways to understand the effect of $T$ without memorizing its value on every single element of $V$. The kernel serves as a sort of abbreviation that encodes the fibers of $T$: i.e., which elements in $V$ map to the same element in $W$. If you think of $T$ as labelling elements of $V$ with elements of $W$, the kernel tells us which elements are labelled with $0_W$. It turns out that's enough to tell us whether any two given elements in $V$ have the same label.


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