Corollary 2.2.4: If $Y$ is a closed subspace of a Hilbert space $H$ and $X \subseteq H$, then $(Y^{\perp})^{\perp}$ and $(X^{\perp})^{\perp} = [X]$. Proof: Since $[X]$ is a closed subspace of $H$, and $(X^{\perp})^{\perp} = ([X]^{\perp})^{\perp}$, it suffices to prove only the result concerning $Y$...
Why is $(X^{\perp})^{\perp} = ([X]^{\perp})^{\perp}$?
I've tried to derive this, but I could only derive tautologies.
@AkivaWeinberger your link is a nice one, I read it all a year ago and it give a good overview of neural network, but deep learning is far from being intelligent.
The thing I keep from it is the proof that "neural network can approximate any non-linear function"
Is that intelligence? to be able to find a non-linear function for every task you encounter? I am not sure about that
@user193319 it suffices to prove $X^{\perp}=[X]^{\perp}$. If $A \subset B$, then $B^\perp \subset A^\perp$, so $[X]^{\perp} \subset X^{\perp}$. The other inclusion follows directly from linearity and continuity of the inner product
You do it in such a way that the rate of change of the vector remains perpendicular to the surface (so in a sense, from the surface's perspective there is no rate of change)
The GIF shows parallel transport of two vectors around a loop
I've just approached modal logic reading "An Introduction to Non-Classical Logic" of Graham Priest.
I am looking for some books that treat this argument in a more extensive way than the book I am reading.
I am especially interested in the philosophical side of modal logic.
Can you suggest me s...
ok my bad, I will read about that to try to understand, you mean any close path will have this effect and it is because the curvature of the sphere right?
I still don`t see the vector turning 360° in the case where the vector is orthogonal to "the plan which contain the equator" and the path is the equator. Am I wrong?
Assume you are given a pair of matrices A, B which satisfy AB = BA. Show that if we set $C = A^@ + 2A$ and $D= B^3 + 5I$, then $CD = DC.$ Then try to generalize this in some interesting way, namely find a property for matrices C,D with that certain property, then CD = DC. For example, $C = A^2 +6A$ and $D = 3B^3 - 2I$ will also have CD = DC.
I'm having trouble even showing the first claim. I don't see how the commutativity of A and B implies $A^2B^3 = B^3A^2,$ for example (when you expand it out).
@kyle: So if you think inductively, you should see that if $AB=BA$, then $A^2B=BA^2$ and $AB^2=B^2A$ can be checked easily. Then you should see that if you've proved $A^jB^k=B^kA^j$ holds when $j,k\le m$, then you can see that it holds when $j,k\le m+1$.
@TobiasKildetoft ah, unfortunately I know nothing about Lie algebras, they're one of those things I keep hearing about and I'll need to learn about at some point
I think anything that ends up yielding a bunch of terms with the form $A^{n}B^{m}$ that we can legally swap around by that induction proof (the rest we can deal with by associativity of addition which is easy).
@AkivaWeinberger when you say "perspective of someone" what does that mean, I think my problem is that my view is stuck with the euclidean frame of reference
@Leaky: I'm assuming that you're talking about $z=z(x,t)$, for example. I do not like it when there is a composition involved. That leads to all the Leibniz headaches and misconceptions.
@AkivaWeinberger ok so the vector is gonna point toward the north pole and thus it will rotate while moving on the equator, but still in 3d Cartesian Euclidean space it does not rotate, I need to investigate ^^
What happens when you parallel transport along a line of latitude (not the equator)? Hint: draw a cone that's tangent to the sphere along the line of latitude @rapasite
@AkivaWeinberger My students typically had a hard time understanding why parallel translation in the cone was the same. Of course, I assigned this as a required exercise every year, and did talk about it in class.
@TedShifrin i mean u know i don’t disagree that our president is incredibly dumb but the forces that decide whether or not we want to go to war for whatever incentive is way more than what goes on in the white house
I've just approached modal logic reading "An Introduction to Non-Classical Logic" of Graham Priest.
I am looking for some books that treat this argument in a more extensive way than the book I am reading.
I am especially interested in the philosophical side of modal logic.
Can you suggest me s...
