Questions and Answers
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.
False
A linear system can have exactly two solutions.
False
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.
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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.
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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.
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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.
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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.
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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.
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A 6 x 3 matrix has six rows.
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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.
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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.
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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.
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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.
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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.
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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.
