vector

A vector is anything you can do two things to: add it to another vector, and scale it by a number. Those two operations are what make something a vector, and almost everything in linear algebra is built on them. The most useful starting picture is an arrow: it has a length (how big) and a direction (which way it points), so the arrow from your front door to the mailbox is a vector — '20 steps, northeast'.

The second picture is just as important: a vector is also an ordered list of numbers, like (3, 1) or (2, -5, 0). Each number is a coordinate, telling you how far to go along each axis. These two views — arrow and list of numbers — are really the same thing, and learning to switch between them freely is half of what linear algebra teaches. The arrow gives geometric intuition; the list of numbers is what you actually compute with.

A tiny example: the vector v = (3, 1) is the arrow that goes 3 steps right and 1 step up. Adding vectors means adding matching coordinates, so (3, 1) + (1, 2) = (4, 3) — geometrically, lay the second arrow at the tip of the first and see where you land. Vectors do not have to be 2D: a list of 100 numbers is a perfectly good vector living in 100-dimensional space.