exzebachay (exzebachay) wrote in diffeq,

Help! Linear Algebra

We're covering Images and Kernels in my Diff Eq class, and I'm having some problems with images of linear transformations.

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?
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