1.2 Row Reduction and Echelon Forms Flashcards
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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.

<p>False</p> 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.

<p>False</p> Signup and view all the answers

What must be true of a linear system for it to have a unique solution?

<p>The system is consistent and the system has no free variables.</p> Signup and view all the answers

Can a system of linear equations with fewer equations than unknowns have a unique solution?

<p>No, it cannot have a unique solution.</p> 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?

<p>There is a pivot position in each row of the coefficient matrix, preventing inconsistency.</p> Signup and view all the answers

Is the system consistent if its 3x5 augmented matrix fifth column is not a pivot column?

<p>Yes, it is consistent.</p> Signup and view all the answers

In the described augmented matrix, is the rightmost column a pivot column?

<p>False</p> 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?

<p>False</p> Signup and view all the answers

Therefore, by the Existence and Uniqueness Theorem, is the linear system consistent?

<p>Yes, it is consistent.</p> 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.

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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.

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