Restate the last sentence in Theorem 2 using the concept of pivot columns: “If a linear system is consistent, then the solution is unique if and only if ________________”

A pivot column is a column that contains a pivot.

If a linear system is consistent, then the solution set contains either (i) a unique solution when there are no free variables or (ii) infinitely many solutions when there is at least one free variable.

When the coefficient matrix of the linear system contains a pivot in each column, its row echelon form of the augmented matrix looks like the upper triangular matrix with nonzero diagonal entries.

This implies that there is no free variable. So, the solution is unique. Here, it is simply known that each column in the coefficient matrix is a pivot column.

Suppose the number of pivot columns is less than the number of variables, then there should be at least one free variable that occurs. This implies that the solution set contains infinitely many solutions.

Hence, we can restate the last sentence in theorem 2 as “If a linear system is consistent, then the solution is unique if and only if every column in the coefficient matrix of the linear system is a pivot column; otherwise, there are infinitely many solutions.”