Questions and Answers
In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form.
False
The row reduction algorithm applies only to augmented matrices for a linear system.
False
A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.
True
Finding a parametric description of the solution set of a linear system is the same as solving the system.
Signup and view all the answers
If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated linear system is inconsistent.
Signup and view all the answers
What must be true of a linear system for it to have a unique solution?
Signup and view all the answers
Can a system of linear equations with fewer equations than unknowns have a unique solution?
Signup and view all the answers
For a system with a 5x7 coefficient matrix to be consistent, what must be true if it has five pivot columns?
Signup and view all the answers
Is the system consistent if its 3x5 augmented matrix fifth column is not a pivot column?
Signup and view all the answers
In the described augmented matrix, is the rightmost column a pivot column?
Signup and view all the answers
In the echelon form of the augmented matrix, is there a row of the form [0 0 0 0 b] with b nonzero?
Signup and view all the answers
Therefore, by the Existence and Uniqueness Theorem, is the linear system consistent?
Signup and view all the answers
Study Notes
Row Reduction and Echelon Forms
- Each matrix corresponds to one unique reduced echelon matrix; multiple sequences of row operations cannot yield different reduced forms.
- The row reduction algorithm is applicable to any matrix, not just augmented matrices associated with linear systems.
Basic Variables
- A basic variable is defined as one that corresponds to a pivot column in the coefficient matrix of a linear system.
Solution Sets
- Expressing the solution set of a linear system through a parametric description only applies if the system has at least one solution.
- A linear system with an echelon form row of [0 0 0 5 0] does not imply inconsistency, as it leads to a valid equation.
Conditions for Unique Solutions
- For a linear system to have a unique solution, it must be consistent and have no free variables.
- An underdetermined system, where the number of equations is fewer than the number of unknowns, cannot have a unique solution due to the presence of free variables.
Consistency of Systems
- A coefficient matrix with five pivot columns means that there is a pivot position in every row, leading to a consistent system.
- The consistency of a system of equations can be determined by examining if the rightmost column of the augmented matrix is a pivot column.
Existence and Uniqueness Theorem
- A linear system remains consistent if the rightmost column of its augmented matrix is not a pivot column, ensuring no contradictory equations exist.
- In echelon form, a row of [0 0 0 0 b] with b nonzero confirms inconsistency, but if absent, the system is consistent according to the theorem.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of row reduction and echelon forms with these flashcards. Each card presents a statement related to matrix operations, requiring you to determine if the statement is true or false. Perfect for students mastering linear algebra concepts.