Posts tagged with **maths**

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 `|||||||||||`

.

In many showers there are two taps:`Hot`

and`Cold.`

You use them to control both the pressure and the temperature of the water, but you do so indirectly: the pressure is controlled by the sum of the (position of the) two taps, while the temperature is controlled by their difference. Thus, the basis you are given:Hot = (1,0) Cold = (0,1)

Isn’t the basis you want:Pressure = (1,1) Temperature = (1,−1)

( would be the basis isomorphism)

other bases:

- L/R channels (mono, stereo)
- any ℝ→ℝ function can be split into odd and even parts
- The bound states of electrons are the eigenfunctions of the hydrogen atom Hamiltonian and its angular momentum, so describing them is changing the basis (from a standard spatial representation to the physically relevant one).
- (i.e., switching from rectangular to polar coördinates sometimes makes things
*much*much easier. Imagine solving the Schrödinger PDE in rectangular coordinates!) - handwriting may have an infinite basis
- pastries basically have a basis of
`{flour, butter, eggs, sugar}`

- faces

(Source: qr.ae)

from *Which Differential Equations are Easy to Solve?* by John Starrett

**Blairthatcher**

by and © Aude Oliva & Philippe G. Schyns

A hybrid face presenting

Margaret Thatcher (in low spatial frequency)andTony Blair (in high spatial frequency)…[I]f you … defocus while looking at the pictures, Margaret Thatcher should substitute for Tony Blair ( if this … does not work, step back … until your percepts change).

(Source: cvcl.mit.edu)