Recent content by brunob

Hi there! The question is: if I have to prove that a function is a change of variable it is sufficient to prove that the function is a diffeomorphism? i.e. prove that the function is bijective, differentiable, and its inverse is differentiable? Thanks!

Nice example! So, if the velocity is constant it means that the position x(t) is linear and so y(t) is, and due the way the functions are linked I can say that their graphics are parallels. Let me know if I'm wrong. Thanks!

Hi there! I have the following property: If x(t) is a solution of \left\{ \begin{array}{l} \dot{x} = f(x) \\ x(t_0) = x_0 \end{array} \right. then the function y(t) = x(t+t_0) is a solution of the equation with initial data y(0) = x_0 . How could it be interpreted geometrically? Thanks!

Homework Statement I have a function f:M_{n×n} \to M_{n×n} / f(X) = X^2. The questions Is valid the inverse function theorem for the identity matrix? It talks about the Jacobian at the identity, but I have no idea how get a Jacobian of that function. Can I see the matrices as vectors and...

Thanks for your response MisterX. What I mean with "a new dot product" is defining a product between vectors that represents the matrix multiplication. Uhmm, yes that's it, it's an extra exercice they give me in my calculus II course. Why you say it's false?

Hi there! I'm back again with functions over matrices. I have a function f : M_{n\times n} \to M_{n\times n} / f(X) = X^2. Is valid the inverse function theorem for the Id matrix? It talks about the Jacobian at the Id, but I have no idea how get a Jacobian of that function. Can I see that...

Great, got it! I'll write it. Thank you so much!

The determinant of M_\epsilon is always nonzero. Unless M = \begin{pmatrix} -\epsilon & 0\\ 0 & -\epsilon \end{pmatrix}. Is \epsilon fixed or just a number decreasing to zero?

Sorry, you're right it's not the matrices with determinant = 0, I should say nonzero determinant. Thanks!

Hi there! How can I prove that the space of matrices (2x2) nonzero determinant is dense in the space of matrices (2x2) ? I've already proved that it's an open set. Thanks. PD: Sorry about the mistake in the title.

Sorry but I don't understand how you get that, could you explain me clearly how you did it? Thanks.

Ok, got it! Any idea for the differentiability? Thank you so much.

I made this proof about \|g(B)\| does not increase to infinit: \|g(A) -g(B)\| decreases to zero and it's positive. \|g(A) -g(B)\| \leq \|g(A)\| - \|g(B)\| \|g(A)\| is constant because A was fixed, \|g(A)\| - \|g(B)\| is bounded by 0 and \|g(B)\| is positive so it can't increase to infinit...

Thanks for your answer! It seems to be easier than I thought. I don't understand what you did in the last step here: I don't realize how you get that inequation. Also, I don't understand what you mean with this: One more question: in the proof you are assuming that ƒ is the identity map? and...

Hi there! How can I prove that a function which takes an nxn matrix and returns that matrix cubed is a continuous function? Also, how can I analyze if the function is differenciable or not? About the continuity I took a generic matrix A and considered the matrix A + h, where h is a real...