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Questions and Answers
A system of one linear equation in two variables is always consistent.
A system of one linear equation in two variables is always consistent.
True
A system of two linear equations in three variables is always consistent.
A system of two linear equations in three variables is always consistent.
False
A linear system can have exactly two solutions.
A linear system can have exactly two solutions.
False
Two systems of linear equations are equivalent when they have the same solution set.
Two systems of linear equations are equivalent when they have the same solution set.
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A consistent system of linear equations can have infinitely many solutions.
A consistent system of linear equations can have infinitely many solutions.
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A homogeneous system of linear equations must have at least one solution.
A homogeneous system of linear equations must have at least one solution.
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A system of linear equations with fewer equations than variables always has at least one solution.
A system of linear equations with fewer equations than variables always has at least one solution.
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If A is an m x n matrix and B is an n x r matrix, then the product AB is an m x r matrix.
If A is an m x n matrix and B is an n x r matrix, then the product AB is an m x r matrix.
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For the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix.
For the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix.
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The system Ax = b is consistent if and only if b can be expressed as a linear combination of the columns of A, where the coefficients of the linear combination are a solution of the system.
The system Ax = b is consistent if and only if b can be expressed as a linear combination of the columns of A, where the coefficients of the linear combination are a solution of the system.
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If a linear system is consistent, then it has infinitely many solutions.
If a linear system is consistent, then it has infinitely many solutions.
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A system of three linear equations in two variables is always inconsistent.
A system of three linear equations in two variables is always inconsistent.
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A 6 x 3 matrix has six rows.
A 6 x 3 matrix has six rows.
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Every matrix is row-equivalent to a matrix in row-echelon form.
Every matrix is row-equivalent to a matrix in row-echelon form.
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If the row-echelon form of the augmented matrix of a system of linear equations contains the row [1 0 0 0 0], then the original system is inconsistent.
If the row-echelon form of the augmented matrix of a system of linear equations contains the row [1 0 0 0 0], then the original system is inconsistent.
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Every matrix has a unique reduced row-echelon form.
Every matrix has a unique reduced row-echelon form.
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A homogeneous system of four linear equations in six variables has infinitely many solutions.
A homogeneous system of four linear equations in six variables has infinitely many solutions.
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A homogeneous system of four linear equations in four variables is always consistent.
A homogeneous system of four linear equations in four variables is always consistent.
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There is only one way to parametrically represent the solution set of a linear equation.
There is only one way to parametrically represent the solution set of a linear equation.
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A 4 x 7 matrix has four columns.
A 4 x 7 matrix has four columns.
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Multiplying a row matrix by a constant is one of the elementary row operations.
Multiplying a row matrix by a constant is one of the elementary row operations.
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The matrix equation Ax = b, where A is the coefficient matrix and x and b are column matrices, can be used to represent a system of linear equations.
The matrix equation Ax = b, where A is the coefficient matrix and x and b are column matrices, can be used to represent a system of linear equations.
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Study Notes
Linear Equations and Consistency
- A single linear equation with two variables is always consistent, guaranteeing at least one solution.
- Two linear equations in three variables may be inconsistent and do not guarantee a solution.
- Linear systems cannot have exactly two solutions; potential outcomes are either none, one, or infinitely many.
Equivalence and Solutions
- Equivalent systems of linear equations share the same solution set.
- A consistent system may provide infinitely many solutions, indicating various potential outcomes.
- Homogeneous systems (equal to zero) always have at least one solution (the trivial solution).
Matrix Multiplication and Definitions
- The product of an m x n matrix (A) and an n x r matrix (B) results in an m x r matrix.
- For matrix multiplication to occur, the number of columns in the first must equal the number of rows in the second.
- A system represented by Ax = b is consistent if b can be formed from the linear combinations of columns from matrix A.
Characteristics of Linear Systems
- A consistent linear system may not always have infinitely many solutions; there can also be a unique solution.
- A group of three linear equations in two variables can still be consistent, indicating potential solutions exist.
- A 6 x 3 matrix contains six rows, showcasing the structure of matrix dimensions effectively.
Row Echelon Forms
- Any matrix can be transformed into its row-echelon form, ensuring compatibility in solving systems of equations.
- The presence of a non-zero row in the row-echelon form of an augmented matrix does not automatically indicate inconsistency.
- Each matrix possesses a unique reduced row-echelon form, aiding in standardization during calculations.
Homogeneous Systems
- A homogeneous system containing four equations and six variables typically results in infinitely many solutions due to the excess variables.
- A homogeneous system with an equal number of equations and variables is guaranteed to be consistent, usually yielding a solution set including the zero vector.
Parametric Representations and Matrix Characteristics
- Multiple methods exist for parametrically representing the solution set of a linear equation; uniqueness is not a standard characteristic.
- A 4 x 7 matrix does not possess four columns; it actually has seven.
- Elementary row operations do not include multiplying a row matrix by a constant unless specified as non-zero.
Matrix Representation of Systems
- Matrix equations of the form Ax = b effectively represent systems of linear equations, encapsulating coefficients and variables in a concise format.
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Description
Test your understanding of the fundamental concepts in Linear Algebra with this true/false quiz based on Chapter 1. Determine whether statements about systems of linear equations are true or false, and reinforce your knowledge of consistency and equivalency in linear systems.
