Linear Transformation: Does T(V) ⊆ W?

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In the discussion about the linear transformation T: V → W, it is clarified that T(V), the range of the transformation, is indeed a subset of W, the codomain. Participants emphasize that this principle applies to all functions, where the range is always a subset of the codomain. Confusion arises regarding the distinction between range and domain, particularly in cases where functions are undefined. The conversation highlights the importance of understanding modern definitions of functions and mappings, moving beyond outdated concepts. Overall, the discussion reinforces foundational concepts in linear algebra and function theory.
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Say I have a linear transformation T:V##\rightarrow##W. Can I necessarily say that T(V)##\subseteq##W?

I feel like T being a linear transformation would make the function behave enough to force things to not be undefined but I can't be certain..
 
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Of course. T(V) is the range which is always a subset of the codomain.
 
Hmm. I see. Thanks! I'm losing my mind.
 
Do you understand that this is true for any function ever? That is how functions are defined, the range is a subset of the codomain.
 
Yea. I messed up my reasoning with the range and the domain. I switched them around thinking that if something was undefined then it wouldn't be in the range. Like if x=0 and f(x)=1/x then 1/0 is not in the range but it is x=0 that is not in the domain.
 
verty said:
Do you understand that this is true for any function ever? That is how functions are defined, the range is a subset of the codomain.


Any mapping ever, really.
 
A David said:
Any mapping ever, really.

I still have the old mindset where every collection is a set and every mapping is a function. Probably this is from reading books not much more recent than the 60's.
 

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