1.2 ROW REDUCTION AND ECHELON FORMS
A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system. Can such a system have a unique solution?
No, it cannot have a unique solution. Because there are more variables than equations, there must be at least one free variable. If the linear system is consistent and there is at least one free variable, the solution set contains infinitely many solutions. If the linear system is inconsistent, there is no solution.
In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations. Is this statement true or false?
The statement is false. Each matrix is row equivalent to one and only one reduced echelon matrix.
The row reduction algorithm applies only to augmented matrices for a linear system. Is this statement true or false?
The statement is false. The algorithm applies to any matrix, whether or not the matrix is viewed as an augmented matrix for a linear system.
If one row in an echelon form of an augmented matrix is [ 0 0 0 5 0] , then the associated linear system is inconsistent. Is this statement true or false?
The statement is false. The indicated row corresponds to the equation 5x(4)=0, which does not by itself make the system inconsistent.
Finding a parametric description of the solution set of a linear system is the same as solving the system. Is this statement true or false?
The statement is false. The solution set of a linear system can only be expressed using a parametric description if the system has at least one solution.
A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix. Is this statement true or false?
The statement is true. It is the definition of a basic variable.
A) Suppose a system of linear equations has a 3x5 augmented matrix whose fifth column is not a pivot column. Is the system consistent? Why or why not?
(Portion of the Existence and Uniqueness Theorem) A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column. That is, if and only if an echelon form of the augmented matrix has no row of the form [0 ... 0 b] with b nonzero.
What must be true of a linear system for it to have a unique solution?
-The system is consistent. -The system has no free variables.
B) In the augmented matrix described above, is the rightmost column a pivot column?
No.
C) In the echelon form of the augmented matrix, is there a row of the form [0 0 0 0 b] with b nonzero?
No.
Suppose a 5x7 coefficient matrix for a system has fivefive pivot columns. Is the system consistent? Why or why not?
There is a pivot position in each row of the coefficient matrix. The augmented matrix will have eight columns and will not have a row of the form [ 0 0 0 0 0 0 0 1], so the system is consistent.
D) Therefore, by the Existence and Uniqueness Theorem, the linear system is
consistent.
