**Can you give a more visual breakdown of the difference between vectors, spans, basis?**

Imagine a 2-dimensional plane. In that plane, imagine two specific points A and B. A **vector** is an arrow that points from A to B. The length of the arrow is the distance between A and B and is also called the **magnitude** of the vector. A **unit vector** , a vector with a length of 1, can be created from the vector by simply dividing by its magnitude and all this unit vector does is simply point from A to B.

Any two vectors, so long as they are not completely parallel to each other, can form a linear combination that can represent any other vector in our 2-dimensional plane. In this situation, we say that these two vectors **span** the 2-dimensional plane. If the two vectors that span the 2-dimensional plane happen to be perpendicular, then we call the set of those two vectors a **basis** .

The process of creating a basis, two perpendicular vectors, from two non-perpendicular vectors that span the 2-dimensional plane is called the **Gram-Schmidt method** .

Suppose now that our two points A and B are located in a 3-dimensional space. A basis that we might have used to describe a vector within the 3-dimensional plane would not be a basis in the 3-dimensional space. This is because our old basis contains no information about the distance along the 3rd dimension which is perpendicular to our original plane. 2-dimensional basis cannot span the 3-dimensional space.