# Linear transformations

In summary, the conversation discusses the relationship between a linear transformation T:V##\rightarrow##W and the possibility of T(V) being a subset of W. The participants also touch upon the concept of range and codomain, clarifying that the range is always a subset of the codomain for any function. They also mention the difference between undefined and not being in the domain, and the fact that this is true for all mappings. The conversation ends with one participant mentioning their old mindset relating to collections and mappings.

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..

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.