September 28, 2011 7 Comments
As I mentioned before, if I have any algebraic object, some interesting things to think about are
- The object itself
- Subobjects of the object
- Structure preserving maps between objects (called homomorphisms)
- Quotient objects
A linear transformation between two vector spaces and (over a field ) is a function such that
- For all ,
- For all , and , .
- by . This is a trivial transformation, or zero transformation.
- by . This is the identity transformation. We write it as .
- More concretely, by
- by . This is an example of a projection. The picture on the wikipedia page is a good one.
The magic of bases:
Just a reminder, I only want to think about finite dimensional vector spaces. While much of this works for infinite dimensional spaces, treatment of such spaces requires care for which I don’t want to put forth the effort. I will try to always specify, but you should just assume I mean finite dimensional spaces if I forget. I will be sure to mention specifically any time I want to talk about infinite dimensional spaces.
So let be a finite dimensional vector space over , and let be a basis for . Then, for any vector , we can write where each . If is a linear transformation
Though this is just a simple application of the rules for a linear transformation, it tells us something interesting. Namely, if I know what does to an entire basis, then I know what does to every vector in ! Tomorrow I’ll talk about notation for this. Spoilers can be found here.