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Posts tagged with linear

If the astronomical observations and other quantities on which the computation of orbits were absolutely correct, the elements also, whether deduced from three or four observations, would be strictly accurate (so far indeed as the motion is supposed to take place exactly according to the laws of Kepler), and, therefore, if other observations were used, they might be confirmed but not corrected.

But since all our measurements and observations are nothing more than approximations to the truth, the same must be true of all calculations resting upon them, and the highest aim of all computations made concerning concrete phenomena must be to approximate, as nearly as practicable, to the truth. But this can be accomplished in no other way than by a suitable combination of more observations than the number absolutely requisite for the determination of the unknown quantities. This problem can only be properly understood when an approximate knowledge of the orbit has been already attained, which is afterwards to be corrected so as to satisfy all the observations in the most accurate manner possible.

Johann Carl Friedrich Gauß, Theoria Motus Corporum Cœlestium in Sectionibus Conicis solem Ambientium, 1809

(translation by C.H. Davis 1963)

(Source: cs.unc.edu)




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

  • thrown in the trash
  • or show up

after the mapping.

image

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.

image

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.

image
linear maps as multiplication
linear mappings -- notice they're ALL straight lines through the origin!

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).

image

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 |||||||||||.




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).

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)





[Karol] Borsuk’s geometric shape theory works well because … any compact metric space can be embedded into the “Hilbert cube” [0,1] × [0,½] × [0,⅓] × [0,¼] × [0,⅕] × [0,⅙] ×  …
A compact metric space is thus an intersection of polyhedral subspaces of n-dimensional cubes …
We relate a category of models A to a category of more realistic objects B which the models approximate. For example polyhedra can approximate smooth shapes in the infinite limit…. In Borsuk’s geometric shape theory, A is the homotopy category of finite polyhedra, and B is the homotopy category of compact metric spaces.

—-Jean-Marc Cordier and Timothy Porter, Shape Theory
(I rearranged their words liberally but the substance is theirs.)
in R do: prod( factorial( 1/ 1:10e4) ) to see the volume of Hilbert’s cube → 0.

[Karol] Borsuk’s geometric shape theory works well because … any compact metric space can be embedded into the “Hilbert cube” [0,1] × [0,½] × [0,⅓] × [0,¼] × [0,⅕] × [0,⅙] ×  …

A compact metric space is thus an intersection of polyhedral subspaces of n-dimensional cubes …

We relate a category of models A to a category of more realistic objects B which the models approximate. For example polyhedra can approximate smooth shapes in the infinite limit…. In Borsuk’s geometric shape theory, A is the homotopy category of finite polyhedra, and B is the homotopy category of compact metric spaces.

—-Jean-Marc Cordier and Timothy Porter, Shape Theory

(I rearranged their words liberally but the substance is theirs.)

in R do: prod( factorial( 1/ 1:10e4) ) to see the volume of Hilbert’s cube → 0.







199 Plays • Download

The Nervous System

  • dissections of live criminals’ brains
  • animal spirits (psychic)
  • neuron νεῦρον is Greek for cord
  • Galen thought the body was networked together by three systems: arteries, veins, and nerves
  • Descartes as the source of the theory of reflexive responses—fire stings hand, νευρώνες tugging on the brain, fluids in the brain tug on some other νευρώνες, and the hand pulls away—automatically.
  • the analogy of a clock (…today we’re much smarter. We think of brains as being like computers, which is definitely not an outgrowth of today’s hot technology!)
  • cogito ergo sumsensation is what’s distinctive about our brains. How could a clock feel something? (Today again, we’re much smarter: we think it’s the ability to reflect on thought—anything with at least one “meta” term in it must be intelligent.)
  • Muscles fire like bombs exploding (a chemical reaction of two mutually combustible elements)—and the fellow who came up with this theory had been spending a lot of time in the battlefield where bombs were the new technology.
  • autonomic, peripheral, central nervous systems
  • Willis, Harvey, Newton
  • What makes nerves transmit information so fast?
  • Galvani’s theory that electricity is only an organic phenomenon. (Hucksters arise!)
  • The theory of the synapse—it’s the connections that matter.
  • The discovery that nerves aren’t continuous connected strings, but rather made up of billions of individual parts.
  • Activation thresholds—a classic and simple non-linear function!

(Source: BBC)




http://ecx.images-amazon.com/images/I/51ohaNaIEGL.jpg
from C. H. Edwards, Jr.
come some examples of linear (vector) spaces less familiar than span{(1,0,0,0,0), (0,1,0,0,0), ..., (0,0,0,0,1)}.

  • The infinite set of functions {1, cos θ, sin θ, ..., cos n • θ, sin n • θ, ...} is orthogonal in 𝓒[−π,+π]. This is the basis for Fourier series.
    http://www.math.harvard.edu/archive/21b_fall_03/fourier/approximation.gif
  • Let 𝓟 denote the vector space of polynomials, with inner product (multiplication) of p,q ∈ 𝓟 given by ∫_1¹ p(x) • q(x) • dx. Applying Gram-Schmidt orthogonalisation gets us within constant factors of the Legendre polynomials 1, x, x²−⅓, x³−⅗x, x⁴−6/7 x²+9/5, ...
    http://www.math.kth.se/math/GRU/2008.2009/SF1625/CMIEL/taylor%20Files/appviewer_010.gif
  • (and, from M A Al-Gwaiz)
    The set of all infinitely-smooth complex-valued functions that map to zero outside a finite interval (i.e., have compact support). These tempered distributions lead to generalised distributions (hyperfunctions) and imprecision on purpose.




Here’s a physically intuitive reason that rotations ↺

Matrix Transform

(which seem circular) are in fact linear maps.

http://image.shutterstock.com/display_pic_with_logo/581935/107480831/stock-photo-one-blue-suitcase-with-wheels-d-render-107480831.jpg

If you have two independent wheels that can only roll straight forward and straight back, it is possible to turn the luggage. By doing both linear maps at once (which is what a matrix
\begin{pmatrix} a \rightsquigarrow a  & | &  a \rightsquigarrow b  & | &  a \rightsquigarrow c \\ \hline b \rightsquigarrow a  & | &  b \rightsquigarrow b  & | &  b \rightsquigarrow c \\ \hline c \rightsquigarrow a  & | &  b \rightsquigarrow c  & | &  c \rightsquigarrow c   \end{pmatrix}
image
or Lie action does) and opposite each other, two straights ↓↑ make a twist ↺.

Or if you could get a car | luggage | segway with split (= independent = disconnected) axles

http://static.ddmcdn.com/gif/jeep-hurricane-layout.gif

to roll the right wheel(s) independently and opposite to the left wheel(s)

http://web.mit.edu/first/segway/comparison.jpg

, then you would spin around in place.





A piecewise linear function in two dimensions (top) and the convex polytopes on which it is linear (bottom).

A piecewise linear function in two dimensions (top) and the convex polytopes on which it is linear (bottom).

(Source: Wikipedia)


hi-res




Slow and steady.

hi-res