Actually, this is more of a general question relating to a homework problem I already did. I was given the initial wavefunction of a particle in an infinite square well:
\Psi(x,0) = Ax if (0 \leq x \leq \frac{a}{2}), and =A(a-x) if (\frac{a}{2} \leq x \leq a)
And of course \Psi(0,0) =...
My textbook states that for operators on complex vector spaces with dimension greater than one, and real vector spaces with dimension greater than two, that there will be invariant subspaces other than {0} and V.
Maybe the book means for a particular operator?
Homework Statement
Prove or give a counterexample: If U is a subspace of V that is invariant under every operator on V, then U = {0} or U = V. Assume that V is finite dimensional.
The attempt at a solution
I really think that I should be able to produce a counterexample, however...
Homework Statement
Give a specific example of an operator T on R^4 such that,
1. dim(nullT) = dim(rangeT) and
2. dim(the intersection of nullT and rangeT) = 1
The attempt at a solution
I know that dim(R^4) = dim(nullT) + dim(rangeT) = 4, so dim(nullT) = dim(rangeT) = 2.
I also...
ok, but can I at least write a basis for null(T) and range(T)? I can't see how to prove this without defining something, because I know I can't prove this by only referring to the finite dimensions of null and range.
Homework Statement
Prove that if there exists a linear map on V whose null space and range are both finite dimensional, then V is finite dimensional.
The attempt at a solution
I *think* the following is true: For all v in V, T(v) is in range(T), otherwise T(v) = 0 which implies v is in...
Gosh, I must be getting sleepy to overlook the importance of n being unique.
So, I can show that each element of V can be written uniquely as a sum of u + n.
Should I also prove U = {au : a is in F} is a subspace of V
n and n' could definitely be different, but I don't think it matters much since they both get mapped to zero.
Is the result of a = a' is enough to prove uniqueness for a direct sum?
if V were finite dimensional then I could say, dim{null(T)} = dim(V) - dim{range(T)}.
But nothing given in the problem statement will let me assume V is finite.
Homework Statement
Suppose that T is a linear map from V to F, where F is either R or C. Prove that if u is an element of V and u is not an element of null(T), then
V = null(T) (direct sum) {au : a is in F}.
2. Relevant information
null(T) is a subspace of V
For all u in V, u is not...
I was wondering if anyone knew anything about epidemic models which take into account the ability of a disease to mutate. Basically I’m curious if there are any existing models which could predict how a rapidly changing disease might affect the progression of an epidemic, or how slower...
Ok, I went into my maple worksheet and chose to export my graph as an .eps, but the file says that it is postscript. I'm also very confused by some of the instructions on the links. For example at
http://amath.colorado.edu/documentation/LaTeX/reference/figures.html
In the 'Only...
I included all those commands, but I still got the same warning and error.
What's the difference between .ps and .eps and why would latex require me to use .eps?