Posts tagged with Taylor series
The modern abstract view is much more interesting but let’s start at the beginning.
Originally vectors were conceived as a force applied at a point.
As in, “That lawn ain’t mowing itself, boy. Now you git over there and apply a continuous stream of vectors to that lawnmower, before I apply a high-magnitude vector to your bee-hind!”
Thanks Galileo, totally gonna get you back, man
The Galilean idea of splitting a point into its
z-coordinate works with vectors as well. “Apply a force that totals 5 foot-pounds / second² in the
x direction and 2 foot-pounds / second² in the
y direction”, for instance.
Therefore, both points and vectors benefit from adding more dimensions to Galileo’s “coordinate system”. Add a
w dimension, a
q dimension, a
ξ dimension — and it’s up to you to determine what those things can mean.
If a vector can be described as (5, 2, 0), then why not make a vector that’s (5, 2, 0, 1.1, 2.2, 19, 0, 0, 0, 3)? And so on.
4th Dimension Plus
So that’s how you get to 4-D vectors, 13-D vectors, and 11,929-D vectors. But the really interesting stuff comes from considering ∞-dimensional vectors. That opens up functional space, and sooooo many things in life are functions.
(Interesting stuff also happens when you make vectors out of things that are not traditionally conceived to be “numbers”. Another post.)
In the most general sense, vectors are things that can be added together. The modern, abstract view includes as vectors:
- 1-D arrays
- linear functionals
- force vectors
- personal preferences
- the flow of heat
- dinosaur tracks
- tying a knot
- economic transactions
- a story or article, in any language
- a poem
- water flowing
- time series
- one instant during an argument
- a curve
- probabilities associated with various outcomes
- marketing data
- statistical observations
- bids and asks
- the quantum numbers of an atom
- one solution of a differential equation
- a polynomial
- a jpeg or bitmap of the Mona Lisa
- a set of instructions
- a dance move
- someone’s signature
- a secret message
- a Taylor series
- a Fourier decomposition
- turning your mattress
(which you’re apparently supposed to do once a season)
- electromagnetic flux
- part of a trajectory
- one wisp of the wind
- states of affairs
- logical propositions
- distortions in a crystal lattice
- a rotation of Rubik’s cube
- neuronal spike-trains — so, thoughts? perceptions?
Things you can do with vectors
Given two vectors, you should be able to take their outer product or their inner product.
The inner product allows you to measure the angle between two vectors. If the inner product makes sense, then the space you are playing in has geometry. (Not all spaces have geometry — some just have topology.)
And — this is weird — if the concept of angle applies, then the concept of length applies as well. Don’t ask me why; the symbols just work that way.
But the “length” of a song (one of my for-instances above) would not be something like 2:43. The magnitude of a song vector would be the total amount of energy in the sound wave | compression wave.
Also, you can do linear algebra on vectors — provided they’re coming out of the same point. Some might say that the ability to do linear algebra on something is what makes a vector.
That can mean different things in different spaces — like maybe you’re superposing wave-forms, or maybe you’re converting bitmap images to JPEG. Or maybe you’re Photoshopping an existing JPEG. Oh, man, Photoshop is so math-y.
Shearing the mona lisa (linear algebra on an image — from the Wikipedia page on eigenvectors, one of which is the red arrow)
The chief triumph of differential calculus is this:
Any nonlinear function can be approximated by a linear function.
(OK…pretty much any nonlinear function.) That approximation is the differential, aka the tangent line, aka the best affine approximation. It is valid only around a small area but that’s good enough. Because small areas can be put together to make big areas. And short lines can make nonlinear* curves.
In other words, zoom in on a function enough and it looks like a simple line. Even when the zoomed-out picture is shaky, wiggly, jumpy, scrawly, volatile, or intermittently-volatile-and-not-volatile:
Moreover, calculus says how far off those linear approximations are. So you know how tiny the straight, flat puzzle pieces should be to look like a curve when put together. That kind of advice is good enough to engineer with.
It’s surprising that you can break things down like that, because nonlinear functions can get really, really intricate. The world is, like, complicated.
So it’s reassuring to know that ideas that are built up from counting & grouping rocks on the ground, and drawing lines & circles in the sand, are in principle capable of describing ocean currents, architecture, finance, computers, mechanics, earthquakes, electronics, physics.
(OK, there are other reasons to be less optimistic.)
* What’s so terrible about nonlinear functions anyway? They’re not terrible, they’re terribly interesting. It’s just nearly impossible to generally, completely and totally solve nonlinear problems.
But lines are doable. You can project lines outward. You can solve systems of linear equations with the tap of a computer. So if it’s possible to decompose nonlinear things into linear pieces, you’re money.
Two more findings from calculus.
- One can get closer to the nonlinear truth even faster by using polynomials. Put another way, the simple operations of + and ×, taught in elementary school, are good enough to do pretty much anything, so long as you do + and × enough times.
- One can also get arbitrarily truthy using trig functions. You may not remember sin & cos but they are dead simple. More later on the sexy things you can do with them (Fourier decomposition).