Linear Equations Chapter 1 Flashcards

Questions and Answers

What is a linear equation?

Any equation that can be written in the form of ax + by = c, where a, b, and c are constants.

What is a system of linear equations?

A collection of one or more linear equations involving the same variables.

What is a solution of a linear equation?

A list of numbers that satisfies a system of linear equations.

What is a solution set?

The set of all possible solutions of the linear system. Signup and view all the answers

What makes two linear systems equivalent?

If they have the same solution set. Signup and view all the answers

What is a consistent linear system?

A system that has at least one solution. Signup and view all the answers

What is an inconsistent linear system?

A system that has no solutions. Signup and view all the answers

What is a matrix?

A rectangular array of numbers. Signup and view all the answers

What is the size of a matrix?

An m x n matrix is a rectangular array of numbers with m rows and n columns. Signup and view all the answers

What are the elementary row operations?

Replacement, interchange, and scaling. Signup and view all the answers

Define row equivalent.

Two matrices are row equivalent if one can be transformed into another using elementary row operations. Signup and view all the answers

What are two fundamental questions about a system of linear equations?

Is the system consistent? If it has a solution, is it unique? Signup and view all the answers

What is a nonzero column or row in a matrix?

A row or column that has at least one non-zero term. Signup and view all the answers

What is a leading entry in a matrix?

The left-most nonzero term in a nonzero row. Signup and view all the answers

What are the criteria for a matrix to be in echelon form?

<ol> <li>All nonzero rows must be above all rows with only zeros. 2) Each leading entry of a row is in a column to the right of the leading entry of the row above it. 3) All entries in a column below a leading entry are zeros.</li> </ol> Signup and view all the answers

What are the criteria for a matrix to be in reduced echelon form?

<ol> <li>It needs to be in echelon form. 2) The leading entry in each nonzero row is 1. 3) Each leading 1 is the only nonzero entry in its column.</li> </ol> Signup and view all the answers

True or false: when a matrix is row reduced into another matrix, U, in echelon form, there is only one such U.

False Signup and view all the answers

True or false: when a matrix is row reduced into another matrix, U, in reduced row echelon form, there is only one such U.

True Signup and view all the answers

Define pivot position and pivot column.

A pivot position is a location in a matrix that corresponds to a leading 1 in the reduced echelon form of the matrix. A pivot column is a column that contains a pivot position. Signup and view all the answers

Explain the Row reduction algorithm.

pg. 15 Signup and view all the answers

Define basic variables and free variables.

pg. 18 Signup and view all the answers

What is the Existence and uniqueness theorem?

pg. 21 Signup and view all the answers

How to use row reduction to solve a linear system?

pg. 21 Signup and view all the answers

Suppose a 4 X 7 coefficient matrix for a system of equations has 4 pivots. Is the system consistent? If the system is consistent, how many solutions are there?

The system is consistent, and there is one unique solution. Signup and view all the answers

Study Notes

Linear Equations and Systems

A linear equation can be expressed in standard linear form.
A system of linear equations consists of one or more equations that share common variables.

Solutions of Linear Equations

A solution satisfies a system when substituted, making all equations true.
Solutions can vary: none, one unique solution, or infinitely many.

Solution Sets

A solution set encompasses all possible solutions for a linear system.

Equivalent Linear Systems

Two linear systems are equivalent if they possess the same solution set.

Consistency in Systems

A consistent linear system has at least one solution.
An inconsistent system has no solutions, illustrated by non-intersecting parallel lines.

Matrices

A matrix is a rectangular grid of numbers defined by its rows (m) and columns (n).
Coefficient matrix includes only coefficients of the system; an augmented matrix includes these plus constants from equations.

Elementary Row Operations

Basic row operations include replacement, interchange, and scaling.

Row Equivalence

Two matrices are row equivalent if one can be converted into the other with elementary row operations.

Fundamental Questions

Key inquiries involve determining the consistency of the system and the uniqueness of any solutions present.

Nonzero Entries

Nonzero rows and columns must have at least one non-zero component.

Leading Entries

The leading entry in a matrix row is the first non-zero element from the left.

Echelon Form Criteria

Echelon form requires that nonzero rows be above all-zero rows and each leading entry must be positioned to the right of the one above it, with all entries below being zeros.

Reduced Echelon Form Criteria

Additional conditions for reduced echelon form include having leading entries as 1 and ensuring each leading 1 is the only nonzero entry in its column.

Matrix Uniqueness in Forms

An echelon form matrix (U) is not unique; multiple forms can exist.
However, a matrix in reduced row echelon form (U) is unique.

Pivots

A pivot position denotes where a leading 1 resides in reduced echelon form; a pivot column contains this pivot position.

Row Reduction Algorithm

Essential steps in solving linear systems involve the row reduction algorithm.

Basic and Free Variables

Basic variables correspond to pivot positions, while free variables are not leading variables in any row.

Existence and Uniqueness Theorem

This theorem outlines conditions under which systems have solutions, either unique or infinite.

Solving Linear Systems via Row Reduction

Row reduction can be employed systematically to find solutions to linear equations.

Coefficient Matrix with Pivots

If a (4 \times 7) coefficient matrix has four pivots, the system is consistent with four basic variables and three free variables, confirming potential for unique solutions.

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Test your knowledge on linear equations with these flashcards from Chapter 1 of Linear Algebra. Each card covers fundamental concepts such as linear equations, systems of equations, and solutions. Perfect for reinforcing your understanding!