Well, familiarity depends on your level of involvement. So look for the definition of <rank> in your notes/textbook. You can't find it (though i doubt it), search the internet: planetmath.org, wikipedia.org, wolfram sites offer a wealth of information on linear algebra.
m is the dimension of the vector space V. It's maximum number of linear independent vectors in V.
ok so rank is the max. number of linearly independent rows or columns.
I was trying to manipulate rank(T)=dim(range(T)) im not sure if this is the correct starting point?
Of course not, you have to know that from the beginning of the class on linear algebra. The range is a subset of V obtained by taking all possible images through T of all the elements of U (assumed to be equal to the domain of T). The range is also known as codomain of a linear map/operator.
If U is a linear space and T is a linear operator, one can easily show that range(T) is also a linear space, namely a vector subspace of V.