Questions and Answers
In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.
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 left bracket Start 1 By 5 Matrix 1st Row 1st Column 0 2nd Column 0 3rd Column 0 4st Column 5 5st Column 0 EndMatrix right bracket, then the associated linear system is inconsistent.
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The echelon form of a matrix is unique.
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The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process.
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Reducing a matrix to echelon form is called the ______ phase of the row reduction process.
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Whenever a system has free variables, the solution set contains many solutions.
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A general solution of a system is an explicit description of all solutions of the system.
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What does it mean for a linear system to be consistent in relation to the rightmost column of the augmented matrix?
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If a system has a pivot position in every row, why is it considered consistent?
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Can an overdetermined system be consistent? Provide an example.
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Can an underdetermined system have a unique solution?
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Study Notes
Row Reduction and Echelon Forms
- Each matrix has a unique reduced echelon form, regardless of the row operations applied.
- The row reduction algorithm can be applied to any matrix, not just augmented matrices for linear systems.
- A basic variable corresponds to a pivot column in the coefficient matrix.
Solution Sets of Linear Systems
- A parametric description of a solution set is only possible if the system has at least one solution.
- An echelon form row like [0, 0, 0, 5, 0] does not indicate inconsistency in a linear system.
- The echelon form of a matrix is not unique, while its reduced echelon form is unique.
Pivot Positions and Phase of Reduction
- Pivot positions are determined solely by leading entries in the nonzero rows of any echelon form, not by row interchanges.
- Reducing a matrix to echelon form is referred to as the forward phase of the row reduction process.
Free Variables and Solution Existence
- The presence of free variables does not guarantee the existence of solutions; inconsistent systems have no solutions.
- A general solution provides a comprehensive description of all potential solutions for a linear system.
Consistency of Linear Systems
- A linear system is consistent if the rightmost column in its augmented matrix is not a pivot column, avoiding rows of the form [0... 0 b] with b nonzero.
- A system with a pivot position in every row is guaranteed to be consistent.
- Overdetermined systems can be consistent; for example, three equations in two unknowns can share a solution like (2, 4).
- Underdetermined systems cannot have a unique solution, as any consistent system with free variables will contain infinitely many solutions, whereas inconsistent systems have no solutions.
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Test your knowledge with flashcards from GaTech's Linear Algebra Midterm 1. Each card presents a statement about matrices and row operations, challenging your understanding of the concepts. Perfect for review and preparation before the exam.
