The pivot columns in an augmented matrix correspond to the bound variables in the system of equations ( below). The remaining variables are called free variables ( below).
Don’t forget to correctly express the solution set of a linear system. Systems with zero or one solutions may be written by listing their elements, while systems with infinitely-many solutions may be written using set-builder notation.
In each case, solve the system you have created. Conjecture a relationship between the number of free variables and the type of solution set that can be obtained from a given system.
For each of the following, decide if it’s true or false. If you think it’s true, can we construct a proof? If you think it’s false, can we find a counterexample?
If the coefficient matrix of a system of linear equations has a pivot in the rightmost column, then the system is inconsistent.
If a system of equations has two equations and four unknowns, then it must be consistent.
If a system of equations having four equations and three unknowns is consistent, then the solution is unique.
Suppose that a linear system has four equations and four unknowns and that the coefficient matrix has four pivots. Then the linear system is consistent and has a unique solution.
Suppose that a linear system has five equations and three unknowns and that the coefficient matrix has a pivot in every column. Then the linear system is consistent and has a unique solution.