Are these two linear maps equivalent?

The image of T' is equal to the image of TS, and therefore the ranks of both transformations are also equal. In summary, the linear maps S and T can be composed to create a new linear map TS, and the map T' is equivalent to TS, with the same image and rank. The notation Im(A) is not commonly used, and the correct notation for the image of a linear transformation S is Im(S) or S(A).
  • #1
roman93
14
0
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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  • #3
My guess would be imaginary part.
 
  • #4
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) ?
 
  • #5
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.
 

FAQ: Are these two linear maps equivalent?

What does it mean for two linear maps to be equivalent?

Two linear maps are equivalent if they have the same domain and codomain, and if they map the same input vectors to the same output vectors.

How can I determine if two linear maps are equivalent?

You can determine if two linear maps are equivalent by comparing their matrices or by testing if they satisfy the definition of equivalence.

Can two linear maps with different matrices be equivalent?

Yes, two linear maps can have different matrices but still be equivalent as long as they have the same domain and codomain and map the same input vectors to the same output vectors.

What is the importance of equivalent linear maps?

Equivalent linear maps are important in linear algebra because they represent different algebraic representations of the same linear transformation. This allows for easier manipulation and understanding of linear maps.

Can two equivalent linear maps have different bases?

Yes, two equivalent linear maps can have different bases. Equivalent linear maps only require the same input-output relationships, not the same basis vectors.

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