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.
Signup and view all the answers
A consistent system of linear equations can have infinitely many solutions.
Signup and view all the answers
A homogeneous system of linear equations must have at least one solution.
Signup and view all the answers
A system of linear equations with fewer equations than variables always has at least one solution.
Signup and view all the answers
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.
Signup and view all the answers
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.
Signup and view all the answers
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.
Signup and view all the answers
If a linear system is consistent, then it has infinitely many solutions.
Signup and view all the answers
A system of three linear equations in two variables is always inconsistent.
Signup and view all the answers
A 6 x 3 matrix has six rows.
Signup and view all the answers
Every matrix is row-equivalent to a matrix in row-echelon form.
Signup and view all the answers
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.
Signup and view all the answers
Every matrix has a unique reduced row-echelon form.
Signup and view all the answers
A homogeneous system of four linear equations in six variables has infinitely many solutions.
Signup and view all the answers
A homogeneous system of four linear equations in four variables is always consistent.
Signup and view all the answers
There is only one way to parametrically represent the solution set of a linear equation.
Signup and view all the answers
A 4 x 7 matrix has four columns.
Signup and view all the answers
Multiplying a row matrix by a constant is one of the elementary row operations.
Signup and view all the answers
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.
Signup and view all the answers
What is the difference between row-echelon form and reduced row-echelon form?
Signup and view all the answers
What steps are involved in Gaussian elimination?
Signup and view all the answers
What are homogeneous equations?
Signup and view all the answers
When are two matrices considered equal?
Signup and view all the answers
Study Notes
Linear Systems and Consistency
- A system of one linear equation in two variables is always consistent.
- A system of two linear equations in three variables may not be consistent due to potential contradictory equations.
- Linear systems cannot have exactly two solutions; they can have either none, one, or infinitely many solutions.
- Two systems are equivalent if they share the same solution set, regardless of their individual representations.
- Consistent systems may have infinitely many solutions depending on the relationships between the equations.
- Homogeneous systems guarantee at least one solution (the trivial solution), specifically when they consist of linear equations set to zero.
Matrix Operations and Products
- The product of an m x n matrix A and an n x r matrix B results in an m x r matrix.
- Matrix multiplication is defined strictly when the number of columns in the first matrix matches the number of rows in the second.
- A system depicted as Ax = b is consistent if vector b can be formed as a linear combination of the columns of A by utilizing solutions to the system.
Matrix Forms and Characteristics
- A linear system's consistency doesn't always imply infinitely many solutions; it can also have exactly one solution.
- A six-row by three-column matrix is correctly defined with six rows.
- Every matrix can achieve a form known as row-echelon form, with particular properties including the arrangement of zero rows and leading 1's.
- Each matrix possesses a unique reduced row-echelon form, which is a refined version of the row-echelon form where leading 1's are the only non-zero entries in their respective columns.
Types of Linear Equations
- Homogeneous systems that involve more variables than equations typically yield infinitely many solutions, ensuring consistency.
- A homogeneous system with an equal number of equations and variables is guaranteed to be consistent.
- There are multiple ways to parametrize the solution set of a linear equation; uniqueness is not a characteristic of all parameterizations.
Miscellaneous Concepts
- A 4 x 7 matrix has seven columns, not four, highlighting the importance of matrix dimension notation.
- Multiplying a row matrix by a constant is not recognized as an elementary row operation when the constant is not specified to be non-zero.
- The equation Ax = b, involving coefficient matrix A along with column matrices x and b, provides a formal representation of a system of linear equations.
Gaussian Elimination Process
- The Gaussian elimination method involves writing the augmented matrix, transforming it to row-echelon form, and then applying back-substitution to derive the solution.
Equality of Matrices
- Two matrices are considered equal when they match in dimensions and all corresponding entries are identical across both matrices.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of linear algebra concepts with this true or false flashcard quiz focusing on Chapter 1. Challenge your knowledge about the consistency and equivalence of linear systems and their solutions. Perfect for students looking to reinforce their learning in linear algebra.