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What is the inverse of a vector?

mattferraro.dev

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Re: What is the inverse of a vector?

#181

Post by spekcular »

jacobolus wrote: post #171
spekcular wrote: post #158

Earlier quoted context omitted.

Can you give an example of a problem that might appear in an undergraduate physics or math course, whose solution is lengthy and tedious by "usual methods" but dramatically simplified by the use of GA? I have seen examples proposed before and been distinctly unimpressed. Any serious simplifications in solutions are usually due to some notation-agnostic insight.

The nicest written-up example I know is https://www.shapeoperator.com/2016/12/12/sunset-geometry/ * * * As a relatively recent personal example I spent a few months (in bits and pieces) working out a bunch of metrical spherical geometry for myself without reference to past work, with points represented as displacement vectors to stereographically projected points at https://observablehq.com/@jrus/planisphere with the…

That is a strange and unconvincing article. In outline, it goes:

1) Look at this slick solution using geometric algebra.

2) Look at how ugly the trigonometric solution is. GA is so great!

3) And, by the way, one can mechanically translate the GA solution into the usual vector notation.

Point 3 is even a bit understated: GA concepts are really only used for a few lines under "Solving for the Earth's radius." Once you hit the equation for epilson^2, it's just standard algebra and trig.

Anyway, the relevant comparison is not between trig and the GA solution; it's between GA and the usual vector language. It's really the same method in different notation. You take the cross product and separate into parallel and perpendicular components, and then you reach the epsilon^2 equation, and it's the same from there.

Also, I think the author is far too hard on the trigonometric solution. The vector solution is somewhat clever, and it for any given problem that gets placed in front of me, it's not obvious a slick solution exists. On the other hand, the philosophy of trigonometry is that given a completely determined problem about triangles, you can just trig-bash mechanically to get an answer (and here you can even before starting that small-angle approximations will make life easier, so trig is even more attractive). It's really not that bad here. Especially, the comment about it being tricky because one must find a "non-trivial relationship between the four angles" is puzzling. Anyone who's spent time with geometry problems like this knows that the first step is to angle-chase and write in all the value and relations, from which this falls out immediately (and again, totally mechanically). Then you just turn the algebra crank and win.

[I don't have time to think about the spherical geometry stuff. Sorry!]

Re: What is the inverse of a vector?

#182

Post by jabl »

jacobolus wrote: post #180
spekcular wrote: post #167

Earlier quoted context omitted.

> the typical course is still not that much different 15 years later. For context, I checked your profile to see where you did your undergraduate degree. I am familiar with the way calculus is currently taught at that university, and it looks quite similar to the "radical [...] era-appropriate" textbook that you linked (at least based on a quick read of a few chapters). Those courses are also taught in a quasi-active…

I’m glad to hear that. I never interacted with the intro calculus course there. My impression is that most intro calculus courses around the US today still use some book like Stewart, Larson, or Thomas, and still teach in traditional lecture style. In poking around I am also glad to see they switched from Griffiths’s to Townsend’s book for intro QM. Much more conceptually clear with less focus on mindless computation…

> I wonder if anything similar can be done for the undergrad electrodynamics course, which was more or less an experiment of “how many gnarly multiple integrals can you grind before burning out?”

The classical field theory course was one of my favorites at the master level. Classical EM is beautiful in the sense that by sprinkling some math magic you can basically calculate everything from a few basic laws. Everything sort of fits together in a coherent tight package. TBH, the class did have a fearsome reputation for being math-heavy, and many of my class mates struggled (which was weird, because I was never super-strong in math compared to many of them).

Re: What is the inverse of a vector?

#183

Post by Grustaf »

thaumasiotes wrote: post #179
monktastic1 wrote: post #122

Earlier quoted context omitted.

> The standard inner product is of course also an exceptionally typical way to multiply vectors, but the concept of an inverse there doesn't make much sense. Not all vector spaces are equipped with an inner product. The point is that you can start with some simple axioms and build these more complicated things (inner product spaces, algebras over a field, geometric algebras, etc.).

> Not all vector spaces are equipped with an inner product. Any vector space over a field (usually part of the definition of a vector space) is equipped with the standard inner product, because multiplication and addition are part of the definition of a field.

That’s not how it works. Unless you explicitly give the vector space an inner product, it doesn’t have one.

What you probably mean is that “you can always define an inner product” but that’s a very different statement.

That’s not true either though, the different dimensions in a vector space do bot have to belong to the same field, so you can’t assume you can add them together.

Re: What is the inverse of a vector?

#184

Post by Grustaf »

rsj_hn wrote: post #170
Grustaf wrote: post #168

Earlier quoted context omitted.

That looks pretty much equivalent, and they even have a version where F = E + B, just like in the youtube video. But my question is if this F, or the tensor version for that matter, has any physical meaning?

Yes, F is the electromagnetic field whose laws of motion generate the E/M dynamics. https://en.wikipedia.org/wiki/Electromagnetic_tensor (This assumes you believe a "Field" has a physical meaning.)

It’s pretty easy to think about a normal three dimensional vector field representing the magnetic field, the trouble for me is when you combine the two into one tensor field.

Looking at the link reminds me that this entity is used outside of GA, but it still feels a bit weird. More weird than say complex currents, but maybe it actually isn’t.

Re: What is the inverse of a vector?

#185

Post by jabl »

rsj_hn wrote: post #170
Grustaf wrote: post #168

Earlier quoted context omitted.

That looks pretty much equivalent, and they even have a version where F = E + B, just like in the youtube video. But my question is if this F, or the tensor version for that matter, has any physical meaning?

Yes, F is the electromagnetic field whose laws of motion generate the E/M dynamics. https://en.wikipedia.org/wiki/Electromagnetic_tensor (This assumes you believe a "Field" has a physical meaning.)

I think the point is that when learning EM, it's a lot about developing intuition by visualizing in your head how the E and B fields behave in different situations, and how charges interact with them etc. You know, a lot of holding your right hand in the "physicist handshake" position and twisting it.

Sure, the same information is contained in the F tensor, but it doesn't have a similar straightforward geometric intuition.

Re: What is the inverse of a vector?

#186

Post by thaumasiotes »

Grustaf wrote: post #183
thaumasiotes wrote: post #179

Earlier quoted context omitted.

> Not all vector spaces are equipped with an inner product. Any vector space over a field (usually part of the definition of a vector space) is equipped with the standard inner product, because multiplication and addition are part of the definition of a field.

That’s not how it works. Unless you explicitly give the vector space an inner product, it doesn’t have one. What you probably mean is that “you can always define an inner product” but that’s a very different statement. That’s not true either though, the different dimensions in a vector space do bot have to belong to the same field, so you can’t assume you can add them together.

> Unless you explicitly give the vector space an inner product, it doesn’t have one.

Well, no, not at all.

The inner product is still there. It's still an inner product. The space in which your vectors exist is still an inner product space. You may not care about the inner product, but it doesn't cease to exist when you stop looking at it.

Re: What is the inverse of a vector?

#187

Post by rolandog »

mferraro89 wrote: post #128
codeflo wrote: post #50

The writing is cute and the animations are nice, but none of it makes any sense. I stopped reading at > It is important to remember that bivectors have a certain redundancy built into them in the sense that s a ⃗ ∧ b ⃗ = a ⃗ ∧ s b ⃗ s a ∧ b = a ∧s b . We can write them using 6 numbers or 3 numbers, but they actually convey 5 degrees of freedom. Three (real) numbers have three degrees of freedom, by definition. (And n…

author here. I was mistaken about the 5 degrees of freedom bit. Bivectors have three. I'll fix the text tonight. I'm sorry you wasted ten minutes on my nonsense.

Hey, I thought it was wonderfully written. I'm definitely part of your target audience, as I fail to easily grok some algebraic concepts.

Thanks for taking the time to write it, and for making the notation easy to write and understand.

Re: What is the inverse of a vector?

#188

Post by Grustaf »

thaumasiotes wrote: post #186
Grustaf wrote: post #183

Earlier quoted context omitted.

That’s not how it works. Unless you explicitly give the vector space an inner product, it doesn’t have one. What you probably mean is that “you can always define an inner product” but that’s a very different statement. That’s not true either though, the different dimensions in a vector space do bot have to belong to the same field, so you can’t assume you can add them together.

> Unless you explicitly give the vector space an inner product, it doesn’t have one. Well, no, not at all. The inner product is still there. It's still an inner product. The space in which your vectors exist is still an inner product space. You may not care about the inner product, but it doesn't cease to exist when you stop looking at it.

I really don't understand why you would say this, it's obviously false.

Setting aside the subtler point of what it means to "have" something in mathematics:

clearly only some vector spaces even have the potential to introduce an inner product. Consider F for some random finite field. You can make a vector space from it, but what would the inner product be? Or R x F for that matter, you could never give that an inner product.

That's why the concepts of "vector space" and "inner product space" are separate concepts. Some vector spaces aren't, and could never be, inner product spaces.

Re: What is the inverse of a vector?

#189

Post by monktastic1 »

thaumasiotes wrote: post #186
Grustaf wrote: post #183

Earlier quoted context omitted.

That’s not how it works. Unless you explicitly give the vector space an inner product, it doesn’t have one. What you probably mean is that “you can always define an inner product” but that’s a very different statement. That’s not true either though, the different dimensions in a vector space do bot have to belong to the same field, so you can’t assume you can add them together.

> Unless you explicitly give the vector space an inner product, it doesn’t have one. Well, no, not at all. The inner product is still there. It's still an inner product. The space in which your vectors exist is still an inner product space. You may not care about the inner product, but it doesn't cease to exist when you stop looking at it.

https://math.stackexchange.com/a/247438/161555

Re: What is the inverse of a vector?

#190

Post by freemint »

pvitz wrote: post #109
ReleaseCandidat wrote: post #64

Earlier quoted context omitted.

> There are two origins of CS in German universities: electrical engineering and maths. Ah, I see, thanks for the information. Which university would be an example for the latter?

Seriously, which German university (not "FH") doesn't teach analysis and linear algebra in the first terms? I am really wondering.

TU Ilmenau has Analysis and Linear Algebra but that is only heard by mathematicians. There is a different class called "Math for engineers" which mechanics students, computer science students, ... hear which has different tests and a different focus but still teaches solving linear ODEs and stuff.
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