Vectors & Matricies

Individual numbers, real or complex, describe a wide range of things:

In 1582, Pope Gregory XIII decreed that a period of ten days would be omitted from the month of October; so the fourth day was followed, not by the fifth, but by the fifteenth.
If  z = 1 ± i , then the sequence with general term  an = zn  has the property that  an = 2an–1 – 2an–2 .
The single number p describes the numerical ratio of circumference to diameter in any circle.

In application, when a single number can be used to describe a quantity, we call the quantity a scalar. Physical scalar quantities usually come with appropriate units attached to them: "365 days" or "65 mph" or "32 degrees Farenheit".

The functions that we've worked with in precalculus, of the form  y = f(x) , are scalar functions of a scalar variable – they take single number inputs and produce single number outputs. Plots of these functions have a familiar appearance. They are simple plane curves, crossing any vertical line at most once:

Scalars, and scalar functions, describe much of the world around us. There are many things, however, for which a scalar description just won't do...