Spanning sets

I want to introduce a concept to some who won't be familiar with it, and then use it to illustrate a constructive way of thinking. For those who have had a course in linear algebra, please skip to the second part while ignoring my clumsy mathematical explanation in the first.

Spans and Bases

Points on a two-dimensional plane can be expressed by ordered pairs such as (4,5), where the first number is an x-coordinate and the second a y-coordinate. In three-dimensional space, we can use (4,5,2). This extends up to any number of dimensions (although once we get past three, it becomes hard to think about).

In a plane, we can take two vectors like A (0,1) and B (1,0) and use them to express any point in the plane:

(4,5) = 5 × (0,1) + 4 × (1,0) In fact, there are many different pairs of A and B we can do this with. Consider: (4,5) = 5 × (0,1) + 2 × (2,0) or even: (4,5) = 4.5 × (1,1) - 0.5 × (1,-1)

A and B are called a basis because they can give you any point in the plane, just by multiplying them by some numbers. We could also add C (10,11), but we find it doesn't help much:

(4,5) = 5 × (0,1) + 4 × (1,0) + 0 × (10,11) …because C is not independent; it can be made up of A and B: (10,11) = 11 × (0,1) + 10 × (1,0)

A, B and C together are called a spanning set, because they can also give you every point in the plane. A basis is actually a spanning set with the fewest possible vectors — that is, with no extra vectors (C) that can be made from other vectors in the set (A and B).

If you think a bit, you will also realize that in three-dimensional space, a basis must have three vectors in it, and so on.

Everyday applications

The foregoing will be extremely intuitive to anyone with a math or science background, and should make good sense to anyone who has taken high school math. Next, consider how the concept extends to non-mathematical situations.

Every task one attempts requires certain capabilities. In many cases you need a certain level of aptitude or proficiency, or skilled capability in order to accomplish a task. For example, I can skate, but only very poorly, so I can't play hockey… or if I did, it would definitely be laughable.

Sometimes the innate skill you need is very basic, but you can use a tool to magnify its effect. If I were building a house, for example, I could use a hammer and nails; but it requires almost no additional skill to use a nail gun, and the house would get build much faster. In other cases, tools aren't very helpful; better skates will not make me a better skater.

For any task, each of the required capabilities may be affected by your skill and/or the tools at hand. Sometimes one tool may help you achieve more than one requirement. This analogy (capabilities are dimensions; skills and tools provide vectors) allow one to reflect, upon beginning a task:

• What am I trying to achieve? What task must be accomplished?
• What capabilities are required to accomplish the task?
• Do I have the necessary skills? Can I develop them?
• Are there tools I can use to magnify my capability?

Most people do this unconsciously. The benefit of doing it consciously is that you can begin to ask yourself more nuanced questions. For me, the most important of these is: am I relying on too many tools?

Minimalism and marketing

In responding to this question, two key influences need acknowledgment.

One is consumer culture. The function of marketing is to convince you that the product marketed addresses some need or is a tool to help you obtain some capability you require. The obvious downside of relying on advertising to define your requirements is that it will always — indeed, it exists solely to — convince you that the product is necessary. On the other hand, a frank appraisal of your actual needs will lead one to seek out the products which are actually needed, and may save money.

Consider the automobile and the iPod. For too many people, car ownership is the perceived price of entry into society. This is an artifact of marketing — question it and it becomes unclear if or when you will actually require a car of your own. Apple MP3 players are very, very well marketed as the best, indeed the only alternative for portable listening, when in fact myriad options are available.

The other influence is minimalism. Many people refer to themselves proudly as neo-Luddites, as if rejection of technology is preferable to its informed use. No hermit has ever accomplished anything world-changing, and in my experience judicious and careful application of small amounts of technology often yields enormous benefits.

Most people recognize the necessity of constantly building their skillset; this is gospel for career counselors. I don't advocate using only a set of tools that forms a basis for the required capability; but it's always best to consider how large the spanning set must be, and which tools are providing little added benefit while causing needless confusion. The results is a streamlined workflow, more efficient behaviour and less extraneous stuff… points I hope to address in further writing.