Hi I am new to this forum and I would like to say that normally I am very good at math, I actually too Calculus 1 and 2 in High school and would have taken more if I had the time.
Anyways onto my question(s). I have 2.
The first I am trying to decide of the transformation is linear or not. The transformation is
T: F -> R, T(x) = Integral(x'(t)/(t^{2}+1))dt from 0 to 1
I understand that for a transformation to be linear it must satisfy T(x+y) = T(x) + T(y) and T(cx) = cT(x), or closed under addition and scalar multiplication. But I am confused as to where to start on the problem above.
The second problem I am having is finding the basis and dimensions of
U = {all [a_{11} 6a_{12} b_{21 }5a_{22}]}
Once again I would know how to do this if it weren't for the a's and b.
1) After looking to the beginning of the chapter in my book, it states "Let F, as before, be the vector space of all smooth functions" and yes R is the set of all reals.
2) Sorry it seems that I left out some of this problem U = {all [a11 6a12 b21 5a22]} that U is the set of all 2x2 matrices A. A better way to visualize U is...
U = |a 6a|
......|b 5a|
where | are brackets.
I hope this clears some stuff up?
For number 1. It seems F would be the set of differentiable functions from R to R. This is the only way the question would make sense.
The rules of integration and differentiation would tell us that we can pull out a constant, so T(cx(t))=cT(x(t)) is definitely satisfied. The rules of integration also give,
T(x(t)+y(t))=Integral[((x(t)+y(t)))'/(t^2+1)]
=Integral[(x'(t)+y'(t))/(t^2+1)] by the addition rule of differentiation
=Integral[(x'(t)/(t^2+1)] + Integral[y'(t)/(t^2+1)] by the addition rule for integration
=T(x(t)) + T(y(t)) by the definition of T.
So it seems this is a linear transformation. As a side note, I feel that derivatives and /(t^2+1) were just a distraction
Your basis would be what you have written: [1,6;0,5] and [0,0;1,0].
Every element of the set U = {[a,6a;b,5a] : a,b are real} can always be written as a linear combination of the two elements. Since it can be written uniquely in this way, it is a basis. Does that make sense? If W = {[a+b, b; 2a+b, 3a] : a,b are real}, a basis would be {[1,0; 2, 3], [1,1; 1,0]}. The comment you listed above is saying that {[1,0; 0,0], [0,1; 0,0], [0,0; 1,0], [0,0; 0,1]} is a basis for the space of all 2x2 matrices, but there are many more bases for this set.
I wouldn't say that's true. I have just personally never seen several of those symbols used in any of my classes.
So it is in fact my basis, I understand with those 2 matrices I can get my whole set of U, which is what a basis is. It just seems overly simple, or maybe I am thinking that it is harder than it is? I know that I thought the first question I had was harder then it really was.