Linear Transformation: Does T(V) ⊆ W?

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SUMMARY

In the discussion on linear transformations, it is established that for a linear transformation T: V → W, the image T(V) is indeed a subset of the codomain W. This conclusion is based on the fundamental definition of functions, where the range of any function is always a subset of its codomain. The conversation highlights a common misconception regarding the relationship between the range and domain, particularly in cases where functions may be undefined.

PREREQUISITES
  • Understanding of linear transformations in vector spaces
  • Familiarity with the concepts of range and codomain
  • Basic knowledge of function definitions and mappings
  • Awareness of common misconceptions in mathematical reasoning
NEXT STEPS
  • Study the properties of linear transformations in detail
  • Explore the definitions of range and codomain in various mathematical contexts
  • Learn about the implications of undefined values in functions
  • Review modern mathematical literature to update foundational concepts
USEFUL FOR

Mathematics students, educators, and anyone interested in understanding linear transformations and the foundational principles of functions.

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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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