Understanding the meaning of Unique Solution

First, we should deal with the fact that "definitions" cannot be "false". A "conjecture" or "statement" may be true or false. A definition is merely information about how to interpret an undefined phrase in terms of statements that are already defined. So the "definition" of a "system" (of linear equations) should say that it is a set of linear equations in several unknowns. A "solution" to a "system" should also be defined. "Independence" of linear equations must also be defined.

The assertion that "if a system has a unique solution then the equations in it must be independent" is not a definition. It is a statement about things that are already defined. As you showed, it is a false statement. The correct statement is "If a system of linear equations is independent and there are as many equations as unknowns then the system has a unique solution."

It is possible to make definitions that are nonsensical in some way. For example, they might define things that do not exist. If you say "Biggy is defined to be a real number B such that for each real number x, B > x" then you have made a definition. The thing that is defined does not exist. However the definition is not a "false" statement. The statement that "There exists a real number equal to Biggy" is false. That statement is not a definition.

 

I'll have to fix my own statement. I should say: if the set of row vectors that give the coefficients of the unknowns in the equations of the system are an independent set of vectors and there are as many equations as unknowns then the system has a unique solution.