## What is a vector?

So, what exactly is a ‘vector‘?

Like so many other words, it seems to mean something different in different ‘disciplines’.  But, deep down, the meanings are connected through the root of the word, ‘vector.  In Latin, it means ‘carrier’ – so, in all its uses now, it implies a direction, an increase or decrease, or another dynamic component to the basic information it, well, carries.

Recently, I have read a most excellent – clear and understandable – explanation of what it is that a ‘vector’ is.  So, with CodeSlinger’s permission (and a few illustrations and links thrown in by me), without further delay…

### VECTORS: a tutorial by CodeSlinger

The simplest model of a vector is a directed line segment.

On the plane, pick any three points, not lying on the same line.  Call one of them the origin.  Call the line from the origin to the second point a basis vector in the u-direction, and call the line from the origin to the third point a basis vector in the v-direction.

Then you can represent any point on the plane by a sum of appropriately scaled copies these two vectors.

No amount of scaling will turn either of these vectors into the other.  Thus we say they form a basis of the plane, which is what were anticipating when we called them basis vectors in the first place.

Of course, this basis is not unique.  Any two vectors which are not parallel form a basis of the plane in which they lie.

Most of us are used to selecting base vectors which are at a right angle to each other, such as the x-y axis.

However, it always takes two of them, so we say the plane is a  two-dimensional space.

Similarly, we can find triples of vectors which form bases of three-dimensional space, and N-tuples of vectors that form bases of N-dimensional space.

The component vectors are made up of multiples of basis vectors in that dimension: the number of basis vectors defines the number of dimensions of that space.

If a set of vectors forms a basis for a space, then we say that the basis spans the space.  The essential defining characteristic of a basis is that none of the N vectors can be obtained by any combination of scaled copies of the remaining N-1 vectors in the set.

Now, if we choose our basis vectors such that the angle between any pair of them is always 90 degrees, then our basis has the additional benefit that the directions are mutually independent.  No amount of movement parallel to any basis vector results in any movement parallel to any of the others.

When a basis has this property we call it an orthogonal basis.  Going back to our plan , for example, we now have the x and y directions familiar from graphing.

In three-dimensional space, we have x, y and z.  And the idea extends to N-dimensional space, even though there may not be standard names for the basis vectors.  So, now that we have a clear picture of the properties of basis vectors in geometric terms, let’s get a little more abstract.

We can treat any set of N mutually exclusive and jointly exhaustive properties (also called degrees of freedom) as a set of N basis vectors in an N-dimensional representation space.

A representation space is just the set of all possible combinations of these properties.  For example, red, green and blue.  Any colour visible to the human eye can be represented as a sum of appropriately scaled red, green and blue components, but neither red, green nor blue can be obtained by any scaled mixture of the other two.

Thus we say that red, green and blue are the 3 basis vectors of a 3-dimensional colour space.

## From Persia to Iran: a tutorial by CodeSlinger

Iran – and its colourful president Ahmadinejad – are in the news a quite lot these days…

But how many of us really know that much about how Iran became what it is today – a hard-line, fascist theocracy with a decidedly apocalyptic fetish?

While I do know a little bit of their history, my interest in the region kind of waned when they stopped building ziggurats, so I’m a little bit out of touch…  (Aside: soon, I’ll be putting up a video with instructions on how to build a model of a ziggurat, inspired by the Ziggurat of Ur – I’m in the process of preparing kits of it for a class-full of eager grade-5-ers!  Fun!)

In other words, I needed a bit of a tutorial on the more recent (say, 20th century) history of Iran.  CodeSlinger was happy to oblige!

Originally, he posted this as a part (!) of a comment to an earlier post of mine, in which he was answering several of my questions – including What is ‘Cultural Marxism’? (which became a post of its own).

With his permission, here is CodeSlinger’s tutorial on the 20th century events, through which Persia became the Iran of today:

Now that we have all that out of the way , we can see what I mean when I say that the manner in which the Pahlavi Shahs went about modernizing Iran subjected the country to the destructive effects of cultural Marxism.  I’m certainly not saying the Shah of Iran was a Marxist.  I’m pretty sure he was nominally Muslim, though he vigorously pursued the policy of secularization begun by his father, so what they really believed is hard to say.

But I don’t think either of them deliberately set out to harm their country, though the father was clearly the shrewder and more ruthless of the two.  The sense I get from reading about them is that they meant to rule well, if at all possible, but they meant to rule in any case.  The social reforms they introduced were being put into practice everywhere in the modern world at the time, but nowhere had they been in place long enough to allow the tree to be known by its fruit.

The father first appears on the stage of history as Reza Khan, commander of the Persian Cossack Brigade, which he used to seize control of Persia and put and end to the Qajar dynasty in 1923, upon which he became Reza Shah and took the surname Pahlavi.  Being broke, in danger of being swallowed by the Russians, and in danger of being overthrown by the Shiite Imams, Reza Shah implemented a strongly anti-communist police state and gave carte-blanche to the British.

To weaken the Shiites, he mandated European dress for men and supported the so-called Women’s Awakening, which included allowing women to work outside the home and banning the chador (!) in 1931.  Another move calculated to weaken the Imams was finalizing the release of the Jews from the ghettos and repealing restrictions on their entry into the professions.  Anyone in government who seriously opposed him was killed.  In the process, he became one of the richest men in Persia, became loved by the city dwellers but alienated the majority of the population, who were still country folk and devout Muslims, and got into a major confrontation with the Imams.

When he felt strong enough, he turned on the British and broke their stranglehold on the country’s infrastructure.  He cancelled the Anglo-Persian Oil Company’s concession, took control of the currency away from the British Imperial Bank, and nationalized the telegraph system.  He encouraged trade with Germany and Italy to further weaken British and Russian influence.  He also changed the name of the country from Persia to Iran, which means Land of the Aryans in Farsi.  Even so, he declared neutrality when World War II broke out, and allowed neither the Axis nor the Allies to operate on Iranian soil.

Not that it helped him.  In 1941, the British and the Russians, whom he had so far successfully played off against each other, joined forces and occupied Iran — ostensibly because they needed a route by which the Allies could supply war materiel to the Russians, but recouping losses was definitely part of the agenda.  The first thing the British did was force Reza Shah to abdicate in favour of his son, who, they correctly assumed, would be easier to handle.  So Prince Mohammad Reza Pahlavi became Shah of Iran at the age of 22.

In any case, Anglo-Persian Oil Company resumed operations under the new name of Anglo-Iranian Oil Company, and carried on until 1951, when Mohammed Mosaddeq got the Iranian parliament to vote him in as Prime Minister after engineering a coalition that nationalized the company.  In response, Anglo-Iranian pulled all of its people out of Iran and the British navy blockaded the Persian Gulf, which cut off oil revenues and turned Iran into a pressure cooker.

Mosaddeq assumed emergency powers, stripped the Shah of money and authority, and broke off diplomatic relations with Britain.  The Shah fled the country.  All kinds of factions emerged and before long, everybody was stabbing everybody else in the back.  Mosaddeq’s manoeuvrings became increasingly desperate and totalitarian, and this gave the British MI6 what they needed to convince the American CIA that Mosaddeq might get in bed with the communists in a last-ditch effort to keep himself in power.  The CIA mounted Operation Ajax in cooperation with MI6.

To make a long story short, the CIA threw a lot of money around, played everyone against everyone and engineered a coupe that deposed Mosaddeq and put the Shah back on the throne in 1953.  All the gory details of Operation Ajax can be found <a href=”http://web.payk.net/politics/cia-docs/” rel=”nofollow”>here</a>, if you’re interested.  In the end, Anglo-Iranian became British Petroleum, took the lead of a consortium of oil companies, and resumed production.  To consolidate his power, the Shah created a new secret police called SAVAK, whose agents were trained by the CIA and the Israeli Mossad (!) and beefed up the Iranian army, which was funded and equipped by the Americans.  Then he proceeded with his White Revolution in 1963, which we have already touched on.

All of this, of course created the perfect set-up for the backlash that dethroned the Shah for the second and last time in 1979 and put Khomeini firmly in control of Iran.  And for all the reputation that SAVAK had for brutality and torture, its replacement, called VEVAK, has a reputation for being a hundred times worse — of course, not much hard information is available outside Iran, since VEVAK operates without government supervision, but instead answers directly to the Supreme Leader, currently Ayatollah Ali Khamenei — in any case, the stories that are told are perfectly consistent with the methods known to be used by their friends and neighbours, the Taliban.

So, who are the good guys in this story?  I’ll be damned if I can find any.  If I had to pick anybody as the least bad, I guess it would have to be the Shah, but that isn’t saying much.  Not much at all.

However, it’s interesting to note the speculations that the CIA has backed every player in this game since the 1940’s, including Khomeini–!  Why would they do that?  Because it gives them leverage no matter how the balance comes out.  And in the present circumstances, that means leverage to manipulate the level of tension in the region to whatever level they need to set the price of oil where they want it, while justifying whatever level of military presence they deem necessary to keep control of Persian Gulf oil fields out of Russian and Chinese hands.  At the same time, it breeds terrorism, which they can use as a scourge of fear to justify increasingly repressive measures against their own population, back home in America.

As Baron Harkonnen said to Muad D’ib, “there are feints within feints within feints.”