Posts tagged with math

my illustration of the first isomorphism theorem, which says you can replace an arrow `ƒ:X→Y` by a sequence of arrows `surjection ∘ bijection ∘ injection`.

(Source: tjsullivan.org.uk)

Rank-Nullity Theorem

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 `|||||||||||`.

Define the derivative to be the thing that makes the fundamental theorem of calculus work.

roots of `x²⁶•y + x•z⁶ + y¹³•z + x⁹•y¹³ + z²⁶     =   0`

$\dpi{200} \bg_white \large x^{27} \cdot y+x \cdot z^6+y^{13} \cdot z+x^9 \cdot y^{13}+z^{26}$

(Source: imaginary.org)

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)```

($\dpi{200} \bg_white \large \begin{bmatrix}+1 & +1 \\ +1 & -1 \end{bmatrix}$ would be the basis isomorphism)

` `

other bases:

(Source: qr.ae)

Blairthatcher

by and © Aude Oliva & Philippe G. Schyns

A hybrid face presenting Margaret Thatcher (in low spatial frequency) and Tony 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)

`the sine of the reciprocal of [some angle between −1/π and 1/π]`

at increasing resolution

```s <- function(x) sin( 1/x )
plot( s, xlim=c(-1/pi, 1/pi), col=rgb(0,0,0,.7), type = "l", ylab="output", xlab="input", main="compose [multiplicative inverse] with [vertical rect of a circle]" )

```

(Source: amzn.to)