Before starting with this lesson, you should be familiar with vector spaces, especially subspaces and span. 

Definition

Example - A Basis For The Plane

Any two linearly independent vectors form a basis for the plane.  This is illustrated in the following image:

Any point in the plane (that is, any vector) can be written as a linear combination of the two vectors.  In the case above, the indicated point is equal to -2v₁ + 3v₂.

Example - The Standard Basis

Infinite Basis

Crystal Structure

Consider the structure of sodium chloride:

Its cubic structure is strikingly regular, so regular that we can actually form a basis for the coordinate system that it represents.    According to the Wikipedia page on sodium chloride, the lattice constant for this structure is 564.02 picometers in all three directions.  If in the image above, we situate the bottom left hand sodium ion (that little purple one on the right) at (0,0,0), then a suitable basis for the position of unit cells in this crystal lattice is given by the following three vectors { (564.02, 0, 0), (0, 564.02, 0), (0, 0, 564.02)}.  

All that we have done is to take the standard basis {e₁, e₂, e₃} and scale them by the distance between the unit cells.