exzebachay (exzebachay) wrote in diffeq,
exzebachay
exzebachay
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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Heh, I just started Linear Algebra this semester myself. I think your book could not have made that any more confusing. I was taught that you should refer to the set, R^n, as the codomain of the transformation. We were then taught that the range is the set of all values transformed from a given vector by the transformation, L. So, the range is contained within the codomain but not vice versa. Let me offer this interpretation: The codomain is the space the range resides in: (f(x)=x^2 is contained in R^2 and f(x,y,z)=w is contained in R^4) and the range is the set of values created by the transformation.

Now to answer your specific question, the difference between the range and the image is that the range covers all possible values for the vector, x, within the domain. The image refers to a specific set of values of the x vector being transformed. This terminology is tongue tying and I apologize if my description is in any way confusing.

In order to find the image of x under L, for a given vector, x, and a given tranformation, Ax, you simply perform vector multiplication. I hope this helps. If you have a specific problem, I can tell you how it is solved. Good luck. ^^