You can add scalars and vectors! From Zero to Geo 1.11

In this video, I show that contrary to popular belief, you can add scalars and vectors! While this particular idea isn't terribly useful, with a bit of generalization it can lead the way to many useful things, such as multivectors.

Sorry for how long this video took to make! While the final product isn't that long, I rewrote parts of it so many times. I even removed a significant section at the last moment because it was too confusing. I hope that in the end, it was worth it and that this video is informative.

Patreon Supporters:
Christoph Kovacs
David Johnston
Jason Killian
p11
Richard Penner
Rosario
trb

Sections:
00:00 Introduction
00:39 Questions
01:14 Analogy with basis vectors
02:16 Exercise
02:48 Paravectors as a linear space
03:12 Zeros
03:58 General direct sums
05:10 Properties of direct sums
06:11 Conclusion
06:40 Rigorous Construction

[thegiftedspriter]

I really appreciate the inclusion of the rigorous construction at the end. While it can bog down a video to an extent, I think math videos on Youtube could really use a bit more mathematical rigor in general.

[mrcookies]

I always wondered why you could add a real and an imaginary part to construct a complex number. This makes things more clear!

[chilljlt]

when is chapter 2 coming out? So excited!

[linuxp]

Glad to see this series come back. The thing is, at least, for APS (Algebra of Physical Space) and STA (Space-Time Algebra) there are clear matrix representations for every element of the algebra, the scalars being diagonal matrices, equal to the identity scaled by a real factor. Also, their vectors are equal to two sets of 4x4 matrices as in Quantum Mechanics. For anyone new to that. Don't know if any arbitrary algebra has their LA analog, though.

[robharwood]

I also really like how all of this can be *_practically_* implemented (e.g. on a computer, or even with pen and paper) using simple, concrete matrix algebra, using the following 'basis matrices' for 2D geometric algebra.

Note: In the following format, the multivector coordinate-vectors can be read off of the top row-vector of the matrix. If you prefer your multivector coordinate-vectors to be column-vectors, you can just use the transposes of these matrices, and the coordinate-vectors can be read off as the first column-vector of the matrix.

U (= Identity matrix) = 1 (the unit scalar) = the multivector coordinate-row-vector (1, 0, 0, 0) =
[1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1]
(Which happens to be the Identity matrix I(4), but uppercase-letter-I is too easily confused with lowercase-letter-l in sans-serif font in YouTube comments, so I picked U for Unit instead.)

X = e1 (the unit vector in the 1st dimension) = multivetor coordinates (0, 1, 0, 0) =
[0 1 0 0
1 0 0 0
0 0 0 -1
0 0 -1 0]

Y = e2 (the unit vector in the 2nd dimension) = multivector coordinates (0, 0, 1, 0) =
[0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0]

B = e1∧e2 (the unit 2D bivector) = multivector coordinates (0, 0, 0, 1) =
[0 0 0 1
0 0 1 0
0 -1 0 0
-1 0 0 0]
(Which happens to be equal to the matrix-product of XY = (e1)(e2) = geometric-product e1e2 = wedge-product e1∧e2.)

It's easy to show that this works because the matrix-product -- expressed using linear-combinations of these basis-matrices -- results in matrices which represent the geometric-product of their equivalent multi-vectors.

So, for example, the geometric product of the multivectors with coordinate-vectors (1, 2, 3, 4) and (1, 2, -2, -1) is
(1 + 2e1 + 3e2 + 4e1∧e2)(1 + 2e1 - 2e2 - e1∧e2) { geometric algebra notation }
= (1⋅1 + 2⋅e1 + 3⋅e2 + 4⋅e1∧e2)(1⋅1 + 2⋅e1 + (-2)⋅e2 + (-1)⋅e1∧e2) { pull out co-effs }
= (1⋅U + 2⋅X + 3⋅Y + 4⋅B)(1⋅U + 2⋅X + (-2)⋅Y + (-1)⋅B) { substitute in matrices }
= { evaluate each factor as actual matrix }
[1 2 3 4
2 1 4 3
3 -4 1 -2
-4 3 -2 1]
(matrix-multiplication)
[1 2 -2 -1
2 1 -1 -2
-2 1 1 -2
1 -2 -2 1]
= { perform matrix multiplication }
[3 -1 -9 -7
-1 3 -7 -9
-9 7 3 1
7 -9 1 3]
= (3⋅U + (-1)⋅X + (-9)⋅Y + (-7)⋅B) { back to lin. comb. of basis-matrices }
= (3⋅1 + (-1)⋅e1 + (-9)⋅e2 + (-7)⋅e1∧e2) { back to multivector basis }
= (3 - e1- 9e2 - 7e1∧e2) { back to geometric algebra notation }
Which is (3, -1, -9, -7) in multivector coordinate-vector notation.

So, with such a basis, Geometric Algebra has an incredibly *_concrete_* representation as multivectors = matrices, and the general geometric-product = matrix-product.

Of course, vector addition is also just matrix addition.
And the geometric dot-product and wedge-product are easily derived from the general geometric-product (= matrix-product), as has already be shown in other videos.
u ⋅ v = (uv + vu)/2
u ∧ v = (uv - vu)/2
So that, as usual, the geometric-product is expressed simply in terms of geometric dot- and wedge-products:
uv = (u ⋅ v) + (u ∧ v)

[shantonudutta]

The part of rigorous contruction was a nice touch! I consider both intuition and rigour to be important for any maths topic.

[robharwood]

In this video, I finally learned what the Direct Sum of two vector spaces really means and is -- using the example of the context of how it is possible to add 'scalars' and 'vectors' in Geometric Algebra! Nice!

Looking forward to your next videos on multivectors and finally how to multiply vectors. (I've already learned this previously, but I really like your explanations and pedagogical style, so I can't wait to see how *you* show it!)

[ianramos]

is the series continuing? this is fantastic

[Bolpat]

It might be worth mentioning that both the term and the symbol for the direct sum are used for “combining” unrelated linear spaces into one _and_ for the span of the union of subspaces of a linear space with the added information or requirement that any set of vectors (other than zero), one from each space, are linearly independent. That is the equivalent in linear spaces of disjoint unions of sets.

[AlfredDiffer]

Thank you for the section at the end. That settles an issue that has nagged me for years.

[libertytrooper]

Looking forward to the rest of the series because its so far been very useful for me to conceptualize concepts that boggle my mind using other approaches. This really should be an undergraduate course. Thinking about numbers as vectors themselves has led to a couple of 'aha' moments that other approaches had me thinking "yeah, but what if I do this" thoughts.

[caspermadlener]

You can also add an identity to any ring without an identity using this idea.

[Rodrigo-xfoe]

I was a little confuse if U and V can share bases or they are in some form scoped to they own linear space. Thankfully, the rigorous proof make this clear.

[GamingKing-jopy]

just realized with some midnight math that these videos have had an actual background for the past 3 videos.
i like it, looks like inverted paper

[TechJoltd]

I got almost 9 pages of notes now in a notebook I'm using for this. Love the series! Also a high schooler, so kind of like a test audience to see if this stuff makes sense to high schoolers, and so far it is.

[ivlivschan]

Wow, your video publish process is too late and I can't wait to watch the next one! Love from Aisa.

[angeldude]

So the Cartesian product, equipped with addition is the direct sum. Great, so now we have a sum that's also a product! 🙃

[tiagorodrigues]

I was reviewing this series a couple of weeks back, and I was thinking to myself: “this is a great series, but I wonder how much more of LA will we go over in this first preliminary chapter.” It turns out the answer was “not much.” I'm looking forward to getting into the Geometric Product proper! Although I expect you're going to introduce the outer product before getting to the GP...

[evandrofilipe]

So happy to see this in my feed

Edit: hyped for chapter 2

[georgechiporikov]

You can perform Lorentz boosts just within Cl(3, 0) using paravectors, if you have a 4-position r,

r = ct + x

Where ct is a scalar and x is a vector.

If you then have a velocity vector v you can break it up into a scalar rapidity ϕ (hyperbolic angle) and vector direction n:

ϕ = atanh(|v| / c)

n = v / |v|

n^2 = 1 so we can exponentiate it with the hyperbolic version of Euler's formula:

e^ϕn = cosh(ϕ) + n * sinh(ϕ)

Sandwiching r between two of these exponentials gives the Lorentz boosted coordinates r':

r' = e^(-ϕn/2) * r * e^(-ϕn/2)