Recent content by JamesTheBond

What does a "nonsingular derivative" mean. It comes in the following context: "If f: R^2 --> R^2 is a function with a nonsingular derivative everywhere, is f bijective?"

The function is everywhere differentiable and continuous/

Hmm... I can't just use the curves along x and y axes. Because there are bound to be such R^2 -> R functions which elude only those particular axes?

Here is a start of my solution which I am stuck with. Consider this continuous surface f(x,y) = z. Consider arbitrary surface point (i,j,z _i_j). This point must exist (given f(i,j) = z _i_j). Proof by contradiction: Consider continuous functions f(x,j) and f(i,y) which are curves that...

Can you guys help me prove: Given a continuous and differentiable function (or surface) f: R^2 -> R, such that f(x,y) = z ... contour lines can always be drawn... the function is NOT bijective. I've been thinking of choosing any arbitrary point and showing that the curves that intersect to...

Not exactly sure what you mean. How do Jordan blocks get involved?

Why do you guys think that given two 3x3 matrices, they are similar if and only if their characteristic polynomial and minimal polynomial are equal (this reasonably fails for 4v4 matrices though)?

Sorry, I meant to say: for some B_W \subset B_V for some basis of V.

Is it possible to just define g(w') to be 0? It seems too simple.

Ok here is one: If there exists linear functional g on V such that g(\{ w_1, w_2, w_3,..., w_n, v_n_+_1, v_n_+_2, ... , v_r \}) = k_1*w_1 + k_2*w_2 + ... + k_r*v_r where r = dim V and for i=1 to r, k_i, w_i, v_i \in K field This g is I think called a linear form and when it is...

That is of course the basis extension principle (theorem?). So here is my revised take on the Ans (please tell me if I am reasoning it): Take some B_W = { w_1, w_2, w_3, ..., w_n} , n = dim W <= dim V B_W \subset B_V for some Basis of V. We know f(p*w1 + q*w2) = p*f(w1) + q*f(w2)...

I am sorry, I misworded my question. At any rate, what I wanted to know was a confirmation of basis extension. For W < V, B_W (some basis of W) is linearly independent and spans W, therefore, there exists some B_V (basis of V) s.t. B_W < B_V. I believe this is the basis extension theorem...

I am writing a solution for the following problem, I hope someone can correct it, because I am not sure what I am missing. Q. V is a finite dim. vsp over K, and W is a subspace of V. Let f be a linear functional on W. Show that there exists a linear functional g on V s. t. g(w)=f(w). Ans...

Is the basis for the subspace W of a vectorspace V spanned by the basis of the vectorspace V? If so how?

This is a problem was thinking about. If I have a capacitor and fill it with a dialectric of some dialectric constant (the dialectric fits perfectly and is mounted via a frictionless bearing so it can move freely). How fast does it move?