To understand when a system of equations will be consistent in terms of the number of pivot variables
To understand when a system of equations will have a unique solution in terms of the number of pivot variables
To understand how to use the reduced row echelon form of the augmented matrix for a linear system to easily write out the solution set
In class, we came up with statements of the following two theorems:
Theorem1.3.1.Consistency Theorem.
A system of equations is consistent if and only if the row echelon form of its augmented matrix has no pivot entries in the rightmost column. Equivalently, a system of equations is inconsistent if and only if the row echelon form of its augmented matrix has a pivot entry in the rightmost column.
Theorem1.3.2.Uniqueness Theorem.
A system of \(m\) equations with \(n\) variables has a unique solution if and only if its augmented matrix has \(n\) pivot entries and no pivot entry in the rightmost column.
Subsection1.3.1Writing Solution Sets
Activity1.3.1.
For the matrix below, verify that the matrix is in rref (reduced row echelon form) and treat the matrix as an augmented matrix for a system of linear equations. Write out the corresponding system of equations. Use this system of equations to write each variable in terms of just free variables and constants.
Under what conditions would your process for the previous activity not work? In other words, when would it not be possible to write each variable in terms of just free variables and constants.
Subsection1.3.2Determining Consistency/Uniqueness of Solutions
Activity1.3.2.
Give an example matrix that fits each of the following conditions:
A 3 by 4 augmented matrix corresponding to a linear system with a unique solution
A 3 by 4 augmented matrix corresponding to a consistent linear system of equations that does NOT have a unique solution
A 3 by 4 augmented matrix corresponding to an inconsistent system of linear equations
