On the dimension of a linear transformation On the general scope, I was wondering how to define the dimension of a linear transformation, if it even has sense. The only ressource I've found addressing the question is a short youtube video (c.f. https://www.youtube.com/watch?v=Csgbd3n6kmw). 
The video's problem statement goes as such : 
"Let $H$ be a non-zero subspace of $V.$ Suppose $T: V\to W$ is a linear mapping.
Prove $\dim(T(H))=\dim(H)$"
There are many things disturbing me in this video, first of which the dimension of T is only found when applied on a subspace of the domain. But can the dimension of a linear transformation be defined on its whole domain? Like finding dim(T)? 
Also, it is later supposed in the video that T is $1$-to-$1$. So the video's framework is pretty restrictive overall.
Trying to understand what the dimension of a linear transformation could mean, I intuitively associate it to the dimension of its image. I hope you can enlighten me on what it represents. 
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Trying to understand what the dimension of a linear transformation could mean, I intuitively associate it to the dimension of its image.
There is an issue with this statement. The dimension of a transformation is not the question you should be asking, but rather, what is the dimension of the vector spaces.
Suppose there is a linear map $T : V \to W$, with $V$ as the domain and $W$ as the co-domain, satisfying the properties of additivity and homogeneity, and V and W are respective vector spaces over a field $\mathbb{F}$, denoting $\mathbb{R}$ or $\mathbb{C}$
The dimension of the vector space V is, by rank-nullity theorem, the dimension of null(T) (or ker(T)), i.e., all the vectors that get transformed to zero from the domain, plus the dimension of range(T). 
The range is not the dimension of the linear map T.
Can the dimension of a linear transformation be defined on its whole domain?
Again, you must think about exactly what you're asking. In this case, TheTrevTutor chose two vector spaces. 
You have to remember, a linear transformation is map between two vector spaces. In his example, he defined H as a vector space, and defined T(H) as a linear transformation applied to H. The claim is that the dimensions of the vector spaces are the same. You can't find the dimension of the transformation, that makes no sense. The linear map has no such property.
