Both rank and image are different concepts.
,it's a scalar. is vector.
If , it mean .After linear transformation, dimension may be decreased.
Let and be K-vectorial spaces and a linear transformation.
From my notes, we call the rank of as .
And in the exercises I have there is no mention of the rank of a linear transformation. However I've to deal with the image of a linear transformation in the exercises and I've no mention of it in my notes. So my question is "does the rank and the image of a linear transformation is in fact the same thing?".
Ok. I'm confused now. Reading my notes (the professor's ones that is), it reads "proposition : Let and be K- vector spaces, a linear transformation. Hence is a subspace of and is a subspace of ."
Also according to his definition, the rank is a vector. Unless "rango" (in Spanish) doesn't mean "rank" in this specific case, but I really doubt it.
I forgot to precise but in my first post and are vectors, not scalars.