From our book:
"Suppose that L:R^m --> R^n is a linear transformation. The vector space R^m is called the domain and the vector space R^n is called the range of L. This terminology is just the standard terminology that one uses when working with functions. Notice that if v is a vector in the range of f that there is no guarantee that there is a vector u in the domain of f for which f(u)=v. For example if L:(x,y) --> (x+y+x, 2x, 0) is a linear transformation from R^2 to R^3, then there is no vector u in R^2 for which f(u)=(0,0,1). The phenomenon also occurs when studying functions of a single variable; for example if f(x)=x^2, then there is no x=a for which f(a)=-2. The definition below makes a distinction between a set that merely contains all possible outputs of a function and the set that actually equals all of these outputs."
What? It specifically states that "v is a vector in the range of f", so how can we even consider that for f(x)=x^2, then there is no x=a for which f(a)=-2? -2 isn't in the range of f...
The book defines image(L) as being a single set that's contained within range(L), but I have no idea how to calculate it.
Can someone give me some pointers?