A vector is a geometric object that has a magnitude and a direction.
Vectors do not have a position. We can represent them anywhere, although we often do with their tail sitting in the center (the origin).
Two vectors can be added by putting them head-to-tail.
A vector can be scaled by a number called a scalar. If this scalar is negative, it flips the vector.
We can represent vectors using a pair of numbers known as their coordinates. The first number indicates the horizontal distance of the vector's tip, while the second number indicates its vertical distance.
Consider î and ĵ, two unit vectors, meaning they have a length of 1. î points to the right, ĵ points upwards.
The coordinates of a vector indicate how much we need to scale î and ĵ in order to obtain our original vector when they are added together.
We say that î and ĵ form a basis. This means that any vector can be expressed as a sum of scaled versions of î and ĵ.
Other vectors than î and ĵ can form a basis. The coordinates of a vector depends on the basis we choose.
We can think about trying out all possible combinations of the basis vectors and putting them together to create a set (a collection of objects). This set is called the span of the basis.