Posts tagged with **vector**

This is trippy, and profound.

**The determinant** — which tells you the change in size after a matrix transformation 𝓜 —** is just an Instance of the** Alternating Multilinear Map.

(Alternating meaning it goes **+ − + − + − + −** ……. Multilinear meaning linear in every term, ceteris paribus:

)

Now we trip. **The inner product** — which tells you the “angle” between 2 things, in a super abstract sense — **is also an instantiation of the** Alternating Multilinear Map.

In conclusion, mathematics proves that **Size is the same** kind of thing **as Angle**

Say whaaaaaat? I’m going to go get high now and watch Koyaanaasqatsi.

Vector fields pervade. I think about them every time I throw a **frisbee in wind**.

In a **social context**, I think about vectors of intent attached to people talking at a party — vectors of flirtation, vectors of eye movement and attention, and more abstract vectors representing jokes, topics of discussion, dance moves, or songs that are playing.

Also when I’m thinking about **international trade** or just the local flows of money in my community, it’s natural to use the vector-field metaphor to “see” the flows.

I also think of **history** (at different scales) using vector fields. **Wars** are like nation-states or soldiers aiming weapon vectors at each other. **Commerce** has many more dimensions since *goods and money are both multi-dimensional*. Ideas and culture also transmit in a vector-field-like way. **Epidemics** — well, there’s a reason mosquitoes are referred to as disease vectors.

**Information flows**, thoughts, internet bits — anything that can be characterised as a vector, you can expand that thought into a more complicated vector-field thought. Turbulent versus laminar flows of ideas and culture? Maybe it wouldn’t deserve a research grant but it’s fun to think about.

There are pretty obvious physical examples of vector fields — **rivers, wind, **geological eroding forces, magnetism, gravity, flying machines, bridge engineering, parachute design, weather patterns, **your entire body** as it does martial arts or dances. Being measurable, these are the source of most of the neat vector-field pictures you can find online.

(Or you find programmatically simple theoretical vector fields like the above: a vector facing **[−y,x]** is attached to every point **(x,y)**. So for instance the point (**3,4)** has a pointer going out **−4** south and **3** east, which equals a total force of **5**.)

The same metaphors and visualisations, though, are open to interpretation as social or economic variables too. For example a profitable **business** is more of a “sink” or **attractor** for 1-D money flows, while a benefactor is a “source”. Likewise a blog that receives lots of links and traffic is a 2-D attractor on the graph of the web — and Google recognises that as PageRank.

I know of at least one paper that tries to best economists’ utility theory models by imagining a person on a 1-D vector field, trying to avoid minus signs and find a path to plus signs in the space.

There is also a game theory connection. Basins of attraction can draw you into a **locally optimal place** that is not** globally optimal**. You can imagine examples in the **evolution** of animals, in company policies or business practices, or in whole economic systems.

On the one hand it may seem frivolous or crackpottical to generalise these concrete physical concepts to the social or psychological. On the other hand — that’s the power of the generality of mathematics!

Vector fields are surfaces or spaces with a vector at each point. That’s the mathematical definition.

Well I thought the outer product was more complicated than this.

An inner product is constructed by multiplying vectors **A** and **B** like **A**ᵀ × **B**. (ᵀ is for turned.) In other words, timesing each **a** guy from **A** by his corresponding **b** guy from **B**.

After summing those products, the result is just one number. In other words the total effect was to convert two length-**n** vectors into just one number. Thus mapping a large space onto a small space, **R**ⁿ→**R**. Hence inner.

Outer product, you just do **A **× **B**ᵀ. That has the effect of filling up a matrix with the contents of every possible multiplicative combination of **a**'s and **b**'s. Which maps a large space onto a much larger space — maybe squared as large, for instance putting two **R**ⁿ vectors together into an **R**ⁿˣⁿ matrix.

No operation was done to consolidate them, rather they were left as individual pieces.

So the inner product gives a “brief” answer (two vectors ↦ a number), and the outer product gives a “longwinded” answer (two vectors ↦ a matrix). Otherwise — procedurally — they are very similar.

(Source: *Wikipedia*)