Are these two linear maps equivalent?

[roman93]

let S: A ->B and T: B -> C be linear maps.
Then
TS : A -> B -> C.
But am I right in thinking that the map T': Im(A) -> C is the same as TS?

If this is wrong, can you explain why please :)

Thanks very much in advance!

 

[micromass]

What is Im(A)?

 

[Whovian]

My guess would be imaginary part.

 

[roman93]

What is Im(A)?

My guess would be imaginary part.

Im(A) is Image(A) or the space that we get when we apply the linear transformation S to A.
Also T' is the same transformation as T but just on a different domain.

I guess what I wrote in OP was wrong, but is it fine to say that Image(T') = Image(TS),
so Rank(T') = Rank(TS) ?

 

[micromass]

Im(A) is Image(A) or the space that we get when we apply the linear transformation S to A.

The correct notation is Im(S) or S(A). The notation Im(A) is not in use.

I guess what I wrote in OP was wrong, but is it fine to say that Image(T') = Image(TS),
so Rank(T') = Rank(TS) ?

That is indeed correct.