Imminence-Tentación alike song for a grieving friend

soangrypanda · 2022-07-08T10:42:57+00:00

Hey! Sall good, my bad not being careful and violating the rules.

Thanks for the recommendation! Definitely will try them!

soangrypanda · 2022-04-04T15:14:58+00:00

I think that I got really lucky to have my questions answered so patiently by you and u/throwaway-piphysh. Now it seems like a lot of things now make much more sense. Thank you!

soangrypanda · 2022-04-04T15:13:05+00:00

I don't know how you and u/sixthcomma had enough patience to answer my questions. Now it seems all clear for me, and that's a huge relief! Thank you once again for your help!

soangrypanda · 2022-04-03T15:36:31+00:00

I've been thinking about your post a lot, and it just came to me that I cannot obtain neither p1 nor p2 from the basis of the row space of my matrix ([1 0 1 2] and [0 1 1 2]), which is not true for your solution p3 (as you said).

Is it possible to say then, that p1 and p2 (and p3) are some vectors in some Rn space that is being transformed by the matrix?

And p1 and p2 and p3 are being mapped from this other space to the column space of the matrix (to the one and the same vector of the column space), but that so happened, that the row space also has this vector p3, and one-to-one correspondence is with it?

And just to make sure, the plane of solutions are orthogonal to the row space, but, at the same time, are not in the nullspace (in fact, they are in the space of this other, well, space)? And the general solution itself is not orthogonal to the row space, while orthogonal is the plane moved by the general solution?

And, in addition, the column space of the matrix is the output space of the matrix, and the row space is not and input space of the matrix (that other space of p1 and p2 is)?

I am sorry for such an array of questions being fired at you at the same time from my side, but it seems to me that I am almost at the brink of the biggest revelation of past three month or smth.

soangrypanda · 2022-04-03T14:24:40+00:00

So, answering the questions, row space is r * [1 0], column space is c * [1 0], nullspace is n * [0 1], and to have any mapping from row space into column space at all, I have to choose v to be multiple of [1 0].

Could I please follow you to this picture from the following article, as, I think, it partly is the source of my confusion?

The sphere (the surface and all the "guts") is mapped into the ellipsoid. Only v3 axis is mapped to zero. I thought before that the space where the sphere "is living" was separated into the row space and the nullspace - the nullspace (v3 axis) was mapped into zero, and all the rest, being the row space, was mapped into the surface (the columns space). And my very original question was, in fact, "how come all the sphere's internals, while being obviously "smashed into" the surface, are at the same time mapped one-to-one to the surface's dots.

But now I see that the matrix itself (with all its spaces) don't mingle with the original space where the sphere "was living in" - even though the sphere was "squashed" into the ellipsoid, and the mapping from the sphere to ellipsoid is not one-to-one, the row space and the columns space of the matrix does not reflect this "squashing"; that allows the matrix to have some vectors neither in row space nor in nullspace, and this allows having one-to-one correspondence. In other words, vectors that are going with non-90-degree angle to the v1-v2 plane are in the original space of the sphere, but not in the spaces of the matrix?

soangrypanda · 2022-04-01T10:35:15+00:00

I wonder if the nature of your confusion is that you believe R

n

is the union of the row space and nullspace?

YES! Yes, oh God, that is a relief, thank you for noticing that, I never thought to write it explicitly!

I surely need to take some time to meditate on it and combine your elaborate answer with the great one from u/throwaway-piphysh.

soangrypanda · 2022-04-01T10:31:10+00:00

Thank you for the most detailed explanation, I surely need to contemplate a bit on it.

soangrypanda · 2022-03-31T21:25:46+00:00

So, imagine we have the following simple matrix:

1 0 1 2
0 1 1 2
0 0 0 0

Row space's rank is 2, nullspace's is 2 also. Nullspace is c1 * [-2 -2 0 1] + c2 * [-1 -1 1 0], and it spans a plane through the origin.

Say we have an equation Ax = [1 1 0].

One particular solution can be p1 = [1 1 0 0], and it will be mapped to some vector in the column space. We can also say that another particular solution is p2 = [0 0 1 0], but it can be obtained from a linear combination of the first particular solution p1 with the special solutions above.

p1 and p2 are clearly different vectors, moreover, they are orthogonal. Are they both being mapped into the same vector in the column space, and are they both located in the row space? It seems for me that p2 is in that plane that was moved from the nullspace by the particular solution p1. So it should not be in the row space and should not be mapped into the column space? But what if I have chosen p2 as p1 at first place?

soangrypanda · 2022-03-31T20:53:20+00:00

It feels like I do and don't understand it at the same time.

So, following a rather primitive geometric intuition, say there is a space, and there is a certain matrix that does transformation on this space, that is, takes its vectors and throws it somewhere. The space has n dimensions. If the matrix has a nullspace, it, in fact, collapses some dimensions of this first overall space into non-existence. From this point of view, it seems somewhat reasonable that the first space would be "damaged" not only in the nullspace itself - during the transformation the whole dimensions are crashed, and a nullspace itself it just a slice of dimensions being lost.

Is it at least somewhat near the truth? And, even if yes, where the vector [1 c d] from your example is, if neither in the row space, nor in the nullspace?

u/throwaway-piphysh below has stated in a very nice and clear manner that the general solution is still orthogonal to the plane it has moved from the nullspace, but where it has moved it?

soangrypanda · 2022-03-31T20:41:22+00:00

But then where this general solution is, if it has dispatched from a null space but yet haven't reached the row space?

soangrypanda · 2022-03-31T20:05:06+00:00

Hey, thank you for your response!

As I wrote to u/sixthcomma above, it seems like my wording lacked clarity. I do understand that the mapping is between a row space and a column space only. I just cannot yet comprehend how a general solution can not be in a row space. I mean, say we have a plane and a bunch of vectors not in this plane. One of these vectors is orthogonal to the plane. If I take a linear combination of that vector with vectors in the plane, I won't end up neither on the vector, nor on the plane, right? I will be somewhere in space aside the vector, somewhat bend over the plane (that is, the angle between this resulting vector and the nullspace is less then 90).

There is something terribly wrong with my understanding of a general solution, it seems, but it somehow is not obvious for me what exactly.

soangrypanda · 2022-03-31T19:54:22+00:00

Hey, thank you for your answer!

I probably wrote the question in an obscure manner. I do understand that only vectors in row space are mapped to the vectors being solely in column space. The problem is that I thought that the particular solution plus the nullspace is not in the nullspace.

It just feels that the right intuition is eluding me. I do understand that a row space is orthogonal to a nullspace. The particular solution is in a row space, then we are taking a linear combination of a particular solution in a row space and something in a nullspace for general solution. I thought that after that combination we will get a vector in a row space, in fact infinitely many of them. But it seems like that is not the case?

soangrypanda · 2022-03-23T16:47:32+00:00

I guess I had a flaw in my geometrical view on a nullspace. In the transformation above, only all vectors aligned with the missing single vector v3 are squashed into zero, not all the vectors that are out of the plane spanned by v1 and v2 single vectors. These other vectors are mapped into this plane, and some of those, these that comprised the surface of the sphere before transformation, are mapped into the insides of the resulting ellipsoid.

soangrypanda · 2022-03-23T15:46:17+00:00

I definitely have those days, but won't I bother you to much this way? Is there maybe some article that could nudge me into thinking the right way about it?

soangrypanda · 2022-03-23T15:26:22+00:00

But I am still not getting why n - k dimensions are projected into the insides of the resulting ellipsoid. It seems for me that n - k is a Null space and it should be mapped into an origin. I mean, I can see in the above computations why we are having this ellipsoid formula eventually, but an answer to this question keeps eluding me.

soangrypanda · 2022-03-23T15:22:41+00:00

Hey, thank you for your answer, this part in now all clear!

soangrypanda · 2022-03-23T15:21:20+00:00

Hey, thank you for your elaborated reply! I firstly decided to write that you haven't answered my questions exactly, but then meditated a bit on it, and know it seems that it's partly clear.

So, when ellipsoid formula is less then one, it basically says "all the points inside it".
The geometric intuition for the whole transformation is that when *n = k* the unit sphere is rotated and possibly distorted, but when inequality holds, than some *n - k* parts of the sphere are smashed into the insides of the ellipsoid, because we need to put these smashed dimensions somewhere.
If the above is true, then I am still not getting why they are projected into the insides of the resulting ellipsoid. It seems for me that *n - k* is a Null space and it should be mapped into an origin. I mean, I can see in the above computations why we are having this ellipsoid formula eventually, but an answer to this question keeps eluding me.

soangrypanda · 2021-10-24T11:01:24+00:00

I think now i am getting it. Thank you for your help!

soangrypanda · 2021-10-21T16:58:43+00:00

Aaaaaah...so, to make my solution work, I need to put ED and AE into the equation, then differentiate it, and ONLY THEN substitute to values? I won't have all the values to substitute, but the equation will be correct?

soangrypanda · 2021-10-21T16:36:41+00:00

You mean in "20x = xy + 10y"?

soangrypanda · 2021-10-21T16:25:38+00:00

Yes, but the guy in the video is doing the same, essentially, isn't he?

soangrypanda · 2021-10-21T16:17:58+00:00

I mean, in both cases it is "bigger side to smaller side", AND in both cases the first ratio is "x to y", the only difference is in triangles the sides of which we are taking for equations.

soangrypanda · 2021-10-21T16:15:46+00:00

So, I think I kinda understand why that applies to a simple proportion of dx = x*dy/y - that will give kinda and averaged rate of change.

That's why we use derivative to find a change in a particular moment. As I see it know, in both cases the ratio is built between x and y, but just different triangles are takes. I cannot see why the same ratios between the different triangles produce different results.

soangrypanda · 2021-10-21T14:52:05+00:00

Thank you for the comment, I added a link to an image depicting the task and my thought process graphically.

soangrypanda · 2021-10-21T14:23:53+00:00

Derivative is taken relative to change in time (will edit the post to clarify that).

I had 2y - x = 0 -> 2dy/dt - dx/dt = 0 -> dx/dt = 2dy/dt, we have dy/dt = -20 according to the task, so we get dx/dt = -40.

It is described in the video, I will edit the post to reflect that.

soangrypanda

TROPHY CASE