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.
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If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated linear system is inconsistent.
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What must be true of a linear system for it to have a unique solution?
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Can a system of linear equations with fewer equations than unknowns have a unique solution?
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For a system with a 5x7 coefficient matrix to be consistent, what must be true if it has five pivot columns?
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Is the system consistent if its 3x5 augmented matrix fifth column is not a pivot column?
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In the described augmented matrix, is the rightmost column a pivot column?
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In the echelon form of the augmented matrix, is there a row of the form [0 0 0 0 b] with b nonzero?
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Therefore, by the Existence and Uniqueness Theorem, is the linear system consistent?
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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.
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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.
