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Thanks! That makes some sense.

I'm probably being a bit dense by the way, I appreciate the help.

Good explanation. But with respect to this statement: "and we construct this surface by finding all the curves with this tangent vector." Won't some curves to this tangent vector be outside the surface? Like if we imagine a plane on a unit sphere with a vector tangent at one point on...

Thanks voko. So in essence we have a tangent vector <dx/ds,dy/ds,dz/ds> that is TANGENT to <a(x,y), b(x,y),c(x,y)> and therefore perpendicular to the unit normal? I feel like I'm still not getting this. Should I know those from systems of ODEs? Should I review that maybe?

Homework Statement How were the integral lines dt/a = dx/b derived from the PDE aUt + bUx = 0 where Ut is the partial derivative with respect to time and Ux with respect to x and a, b are constants. Homework Equations I honestly have no idea. I may be unprepared for this course as...

Also could you point me to somewhere that would prove why this is true?

Do you mean n+1 x n x n-1 etc. ? And don't worry they're finite, thanks for the response.

Homework Statement Count all surjections of A to B, (f: A ---> B) where |A| = |B| + 1 Homework Equations None? This is just a problem I came across online. The Attempt at a Solution I'm really not sure. This isn't technically homework, but I'm just looking for a good method. I...

Ohhh I'm an idiot: Take exactly what I wrote, but with all positive coefficients (i.e. anywhere there is a negative put a positive) and add that to what I wrote. It's equal to <u,v> now isn't it? I think I figured out where I went wrong.

I'm still not sure I understand all the details, but I may just need to think on it more. One other quick question for anyone: -(1/4<u,u> - 1/4<u,v>) + (1/4<v,u> - 1/4<v,v> apparently equals <u,v> (the inner product) - why? Any help would be appreciated, and again thanks to everyone that's...

Also, that use of isomorphism really is cool.

a) suppose dim(V) > dim(W). this means that there are more elements in any basis for V, than there are in any basis for W. pick a basis, any basis, for V, say: {v1,v2,...,vn} consider the set {T(v1),T(v2),...,T(vn)}. can this set be linearly independent? if not, than by the linearity of T...

That's because the Kernel is the subspace of things that get mapped to zero by the transformation right? And injection requires that 0 be the only thing mapped to 0 (i.e. the transformation is linearly independent) and surjection requires that everything in W get mapped to (i.e. transformation...

Yes, since no linear combination of one can make the other. Thanks for guiding me through that question! You don't happen to have time for those other ones up there do you? No worries if you don't. I'm going to post a try at a, b and c above either way!

10-1-1 and 0100? For U and V respectively? I feel a bit more confident about this answer lol, am I wrong?