Posts tagged with **mathematics**

A jet can be thought of as the infinitesimal germ of a section of some bundle or of a map between spaces.

Jets are a coordinate-free version of … Taylor series.

Michael Bächtold, David Corfield, Urs Schreiber

**Pictorial glossary**

Bundle:

Mapping between spaces:

- Mapping a circle onto a line
- Wrapping a line around a circle
- Associating a number to a Euclidean=flat subregion
- Thinking of knots as mappings of a loop 𝕊¹ into ambient space ℝ³.
- Thinking of homotopies as mappings of cubes into spaces.
- Associating photos to geolocations (Flickr, stochasticplanet)

- Mappings from manifold to manifold—for example the derivative (pushforward) at each point of the surface.
- Associating vectors (based force-arrows) to spots on the planet (2-sphere):

*The dual V* of a vector space V over ℝ matches lists of reals to linear functionals.*

What’s the simplest way to say this? Talk about a number like “5”. Initially I think of it as 5 stones ⬤⬤⬤⬤⬤. But I could also imagine a line through the origin with a slope of 5, representing the verb `quintuple`

.

*pictures of lines through the origin with various slopes*

Seen as a function ƒ₅=quintuple, the-line-through-the-origin-with-a-slope-of-5, is `ƒ₅(x)=5•x`

. That ƒ₅ does things like

- ƒ₅(■■■)=5+5+5 and
- ƒ₅(■■■■■■)=5+5+5+5+5+5.

Counting in the dual space ƒ₀,ƒ₁,ƒ₂,… would look like `_ / ∕ ...|`

. Increasing slope from _ to ⁄ to | instead of increasing number from 0 to 1 to ∞. Or I could say `id`

, `double`

, `triple`

, `quadruple`

, `quintuple`

, ….

(Why did I jump so suddenly 0,1,… from _ flat to ⁄ 45°? This just proves that half of the ℝ⁺ are stuffed between [0,1) and the other half are between (1,∞).

To jump between the two worlds you use the reciprocal map flip(■)≝1/■. T

hen you’d be counting id, half, third, fourth, fifth, sixth, seventh… Infinity in a teacup.)

These two things—the five rocks ⬤⬤⬤⬤⬤ and the function ƒ₅—aren’t even the same *kind* of thing. One is nouns and one is a verb.

But still, for any real number that I “counted” I could match up a function, just like I did with

- "5 that I counted" and
- "function ƒ₅ =
`quintuple`

”

So these two qualitatively different things are in bijection. (One can hope for insights by viewing things through one lens or the other, noun or verb version.)

This one-dimensional story can be upgraded to a multi-dimensional one where

- lists of reals
`(3.1, √2, −2.1852, ..., 6)`

match to

- many-to-one functions
`ƒ( list ) = 3.1•list[first] + √2•list[second] − 2.1852•list[third] + ... + 6•list[Nth].`

Translating between the noun and verb viewpoints is then called musical isomorphism, represented with ♭ and ♯ symbols. Raising and lowering indices in a tensor is ♯ and ♭.

The rank-nullity theorem in linear algebra says that dimensions either get

- thrown in the trash
- or show up

after the mapping.

By “the trash” I mean the origin—that black hole of linear algebra, the `/dev/null`

, the ultimate crisscross paper shredder, the ashpile, the **wormhole to void and cancelled oblivion**; that country from whose bourn no traveller ever returns.

The way I think about rank-nullity is this. I start out with all my dimensions lined up—**separated, independent**, not touching each other, not mixing with each other. `|||||||||||||`

like columns in an Excel table. I can think of the dimensions as separable, countable entities like this whenever it’s possible to rejigger the basis to make the dimensions linearly independent.

I prefer to always think about the linear stuff in its preferably jiggered state and treat how to do that as a separate issue.

So you’ve got your `172 row × 81 column`

matrix mapping 172→ separate dimensions into →81 dimensions. I’ll also forget about the fact that some of the resultant →81 dimensions might end up as linear combinations of the input dimensions. Just pretend that each input dimension is getting its own linear λ stretch. Now **linear just means multiplication.**

Linear stretches λ affect the entire dimension the same. They turn a list like `[1 2 3 4 5]`

into `[3 6 9 12 15]`

(λ=3). It couldn’t be into `[10 20 30 − 42856712 50]`

(λ=10 except not everywhere the same stretch=multiplication).

Also remember – everything has to stay centred on ~~0~~. (That’s why you always know there will be a zero subspace.) This is linear, not affine. Things stay in place and basically just stretch (or rotate).

So if my entire 18th input dimension `[… −2 −1 0 1 2 3 4 5 …]`

has to get transformed the same, to `[… −2λ −λ 0 λ 2λ 3λ 4λ 5λ …]`

, then linearity has simplified this large thing full of possibility and data, into something so simple I can basically treat it as a stick `|`

.

If that’s the case—if I can’t put dimensions together but just have to λ stretch them or nothing, and if what happens to an element of the dimension happens to everybody in that dimension exactly equal—then *of course* I can’t stick all the 172→ input dimensions into the →81 dimension output space. `172−81`

of them have to go in the trash. (effectively, λ=0 on those inputs)

So then the rank-nullity theorem, at least in the linear context, has turned the *huge* concept of dimension (try to picture 11-D space again would you mind?) into something as simple as counting to 11 `|||||||||||`

.