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 refering 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.
1. 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...
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?
Latex gave me one warning and one error.
Latex warning: file 'myfile.ps' not found on line41
! Latex error: unknown graphics extention: .ps
I did use \usepackage{graphicx} in my document, I just typed it wrong in this thread.
And as far as having a graphics package instaled, I'm...
hi,
I'm trying to put a graph generated in maple into a latex document, but I have no experience using either program. So far I've been able to save my maple plot in postscript format, and based on various online tutorials I've included the \usepackage{graphics} comand after...
Since u and v are elements of the intersection, u and v will also be elements of any subspace W that is in the intersection. And since u and v are in W and W is a subspace, this guarantees that u+v will also be in W. This same argument would apply to scalar multiplication.
Is that the...
Prove that the intersection of any collection of subspaces of V is a subspace of V.
Ok, I know I need to show that:
1. For all u and v in the intersection, it must imply that u+v is in the intersection, and
2. For all u in the intersection and c in some field, cu must be in the...
1. Homework Statement
Prove: If a, b are nonzero elements in a PID, then there are elements s, t in the domain such that sa + tb = g.c.d.(a,b).
2. Homework Equations
g.c.d.(a,b) = sa + tb if sa + tb is an element of the domain such that,
(i) (sa + tb)|a and (sa + tb)|b and
(ii) If...
1. Homework Statement
Let G_1 and G_2 be groups with normal subgroups H_1 and H_2, respectively. Further, we let \iota_1 : H_1 \rightarrow G_1 and \iota_2 : H_2 \rightarrow G_2 be the injection homomorphisms, and \nu_1 : G_1 \rightarrow G_1/H_1 and \nu_2 : G_2/H_2 be the quotient...
What are you trying to solve? You could substitute numbers for the x's and compute the value.
I don't believe there's a nifty identity for (sin(3x))^2 + (cos(3x))^2 even though it does look somewhat similar to (sin(x))^2 + (cos(x))^2.
Ironically, the book is called 'Linear Algebra Done Right' 2nd ed. by Sheldon Axler. I don't exactly love it, but it is what I'll be using this fall so I better get used to it. :rolleyes:
Thanks for all the help!
The book I'm working from does not discuss infinite dimensional vector spaces. It only gives a brief description of $\mathbf{F}$^{\infty} and P(F), the set of all polynomials with coefficents in $\mathbf{F}$.
In particular it says, "because no list spans P(F)...
Prove that $\mathbf{F}$^{\infty} is infinite dimensional.
$\mathbf{F}$^{\infty} is the vector space consisting of all sequences of elements of $\mathbf{F}$, and $\mathbf{F}$ denotes the real or complex numbers.
I was thinking of showing that no list spans $\mathbf{F}$^{\infty}, which would...
Well, in question B. they ask for the spring force which is in units of newtons.
The units in your answer are in joules, but you need them to be in newtons in order for the answer to make sense.
We don't do that here.
However, if you show the work you've already done on this problem, people will point out your mistakes and help you solve your own problem.