Have you looked up the definitions of any of these things?
I have several related questions about different properties of T:
1. Given T, how do I find T*? Can someone give a simple example?
2. Given T and v (where v is a vector), how do I calculate T*(v)?
What are the best/quickest ways to tell if T is:
Is there any difference between A (a matrix) being normal/self-adjoint and T (a linear transformation) being normal/self-adjoint?
Thanks for any help!
Yes, but they are somewhat hard for me to understand. For example, the section in my book on the Adjoint of a Linear operator lists a lot of different theorems involing T*, but I can't seem to find a clear one that says "Ok, this is how you calculate it." Although, I do remember it involving solving a system.
I also wanted to view this information all in one place so I could paste it into my study guide. Thanks for your help!
ok, what we are talking about is linear transformations of a vector space.
there are 2 main ways of looking at these things:
1) as "abstract algebraic objects" which behave certain rules.
2) as a generalization of familiar actual "things" which we can compute.
in the first way, all you need to do is specify the rule the thing behaves according to. in the second way, you need a FORMULA.
this is "bound up" with the 2 main ways of looking at vector spaces: as abstract algebraic entities (which satisfy the axioms (normally they give you 9 or 10 of them) of a vector space), or as collections of vectors, which we think of as being determined by a coordinate system we can scale.
the concept of basis is the "bridge" between the two: once we pick (name) a basis, we can write down the formulas according to the rules. it works like this:
vector-->apply basis--->linear combination ("coordinates in the basis")
linear map--->apply basis--->matrix
this lets us turn "the abstract thingies" into "expressions with field elements" (numbers).
the important thing to remember is that picking a basis, is a somewhat arbitrary choice. that is, although every matrix is a linear transformation, every linear transformation is only REPRESENTED by a matrix. the linear transformation isn't "really the matrix", it's just that matrix, in a certain language (determined by our choice of basis). pick a different basis, you get a different matrix, even though it is the same linear transformation. this is directly analagous to a vector having different coordinates, in different bases. we usually choose bases to make the formulae "nice", because easier to calculate is better (especially if you're lazy, like me).
vector spaces don't come with "preferred" bases. we can choose whatever basis we like. 4 out of 5 mathematicians surveyed prefer to choose standard basis.