Reduced regression models and tests for linear hypotheses

On a SAS discussion forum, a statistical programmer asked about how to understand the statistics that are displayed when you use the TEST statement in PROC REG (or other SAS regression procedures) to test for linear relationships between regression coefficients. The documentation for the TEST statement in PROC REG explains the F test in terms of a matrix of linear constraints. However, the programmer wanted a simpler explanation. Fortunately, there is an easy way to explain the TEST statement from first principles. The explanation involves running two regression models.