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Chapter 1: Q27E (page 1)

Restate the last sentence in Theorem 2 using the concept of pivot columns: 鈥淚f a linear system is consistent, then the solution is unique if and only if ________________鈥�

Short Answer

Expert verified

The required statement is, 鈥淚f 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.鈥�

Step by step solution

01

Definition of pivot column

A pivot column is a column that contains a pivot.

02

Statement of the last sentence in theorem 2

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.

03

Determine when the unique solution occurs

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.

04

Determine when infinitely many solutions occur

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.

05

Conclusion

Hence, we can restate the last sentence in theorem 2 as 鈥淚f 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.鈥�

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