Are these two linear maps equivalent?

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Discussion Overview

The discussion revolves around the equivalence of two linear maps, specifically examining the relationship between the composition of linear maps and the image of a linear transformation. Participants explore the definitions and implications of these concepts within the context of linear algebra.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant proposes that the map T': Im(A) -> C is equivalent to the composition TS: A -> B -> C.
  • Another participant questions the meaning of Im(A), suggesting it might refer to the imaginary part.
  • A later reply clarifies that Im(A) refers to the image of A, which is the space obtained by applying the linear transformation S to A.
  • Some participants note that T' is essentially the same transformation as T but defined on a different domain.
  • There is a suggestion that while the initial claim may be incorrect, it could be valid to state that Image(T') = Image(TS) and consequently Rank(T') = Rank(TS).
  • Another participant emphasizes that the correct notation should be Im(S) or S(A), arguing that Im(A) is not standard notation.

Areas of Agreement / Disagreement

Participants exhibit some agreement on the definitions of image and rank, but there is disagreement regarding the initial claim about the equivalence of the maps and the appropriate notation. The discussion remains unresolved on whether the original assertion about TS and T' holds true.

Contextual Notes

There are limitations regarding the notation used and the assumptions about the equivalence of the maps, which have not been fully clarified or resolved.

roman93
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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!
 
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What is Im(A)?
 
My guess would be imaginary part.
 
micromass said:
What is Im(A)?

Whovian said:
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) ?
 
roman93 said:
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.
 

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