Vectors

A vector is one idea wearing three costumes, and fluency in linear algebra is really the ability to switch between them without thinking.

To a physicist it is an arrow with a length and a direction. To a programmer it is an ordered list of numbers, [3, 2]. To a mathematician it is simply a thing that can be added and scaled. All three are the same object, and the third definition is the one that matters, because it is the one that lets vectors describe things that are not arrows at all — a colour, a portfolio, a row of exam marks.

When we draw the vector [3, 2] we draw an arrow from the origin to the point three right and two up. That starting point is not a detail — nearly everything in linear algebra assumes vectors begin at the origin, so that a list of numbers and an arrow are the same information written two ways.

The numbers are called the components. In two dimensions there are two, in three dimensions three, and nothing stops us going further: a vector in ten dimensions is just a list of ten numbers, even though we can no longer draw it.

To add two vectors, add their components: [3, 2] + [1, 3] = [4, 5]. Geometrically you place the tail of the second arrow at the tip of the first, and the sum is the arrow from the very start to the very end. Walking three-east-two-north and then one-east-three-north lands you four-east-five-north, whichever leg you take first — which is why vector addition does not care about order.

Multiplying a vector by a number — a scalar — stretches it: 2·[3, 2] = [6, 4] is twice as long, same direction. A scalar between 0 and 1 shrinks it; a negative scalar flips it to point the opposite way. The word "scalar" exists precisely because these numbers scale.

Scaling never changes the line the arrow lies on (except to reverse it). Every scalar multiple of a single non-zero vector sweeps out exactly one line through the origin — a fact we will lean on the moment we define span.

The magnitude (length) of a vector comes straight from Pythagoras: for [3, 4] it is √(3² + 4²) = 5. In any number of dimensions it is the square root of the sum of the squared components.

The dot product multiplies two vectors into a single number: [a, b]·[c, d] = ac + bd. It measures how much two vectors point the same way. It is zero exactly when they are perpendicular, positive when the angle between them is sharp, and negative when it is wide — which makes it the tool for testing angles and, later, for projections.

The point of the two operations is that they combine. A sum of scaled vectors is called a linear combination, and it is the single most important phrase in the course.