The "range" and "image" of a transformation refer to the same thing, right?

Question

I'm hoping this is true because I've been told that "the rank of a transformation is the dimension of its image", and also that "the rank of a transformation is the dimension of its range". This doesn't make sense if range and image aren't the same thing, so I'm just wanting to be completely sure.

Furthermore, does codomain refer to the image of the transformation or to the vector space to which the images of the transformation are brought?

Answer 1

You have correctly observed that usage varies and there is no standard definition.

I have always seen, used, and taught that if $$f:X\to Y$$ then $X$ is the domain of $f$, $Y$ is the range of $f$, and $f(X)$ is the image of $f$.

But I also point out that others use “codomain” for $Y$ and “range” for $f(X)$.

Answer 2

Most definitions I've seen hold that the codomain is the type of value that could come out of a function. For example, a polynomial function has the set of all Reals as the domain and since only real numbers can result from a polynomial function, the codomain is all Reals, too. (You cannot get imaginary numbers as the output of a polynomial function; all coefficients are Real and so is the domain.) However, polynomials, primarily even-degree ones, do NOT use all the Real numbers as outputs. The outputs a particular function actually uses from the set of all Reals is the image, also sometimes called the range. Thus, what could come out of a function is the codomain, but what actually comes out is the image (or range).