Linear Equations and Solutions

Linear Equations

• A linear equation has the form ax₁ + a₂x₂ + ... + aₙxₙ = b, where a₁, a₂, ..., aₙ are coefficients and x₁, x₂, ..., xₙ are variables.
• A system of linear equations is a set of linear equations with the same variables.
• A system of linear equations can be represented by an augmented matrix. This matrix consists of the coefficients from the equations, with an additional column for the constants.
• Example: a₁₁x₁ + a₁₂x₂ + ... + a₁ₙxₙ = b₁
  a₂₁x₁ + a₂₂x₂ + ... + a₂ₙxₙ = b₂ ... aₘ₁x₁ + aₘ₂x₂ + ... + aₘₙxₙ = bₘ
• The solution set of a system of linear equations is the set of values for the variables that satisfy all equations in the system.

Solution to system of Linear Equations

• A solution to a system of linear equations is an ordered n-tuple (x₁, x₂, ..., xₙ) that satisfies all equations.
• There are three possibilities for a solution set: no solutions (inconsistent), one solution, or infinitely many solutions.

Matrix representation

• A matrix is a rectangular array of numbers (coefficients).
• A coefficient matrix is a matrix of coefficients from a system of equations
• An augmented matrix is created by combining the coefficient matrix and the constant terms. The size of a matrix is expressed as m x n (m rows and n columns).

Elementary Row Operations

• These operations, when applied to the rows of a matrix, alter the system but leave the solution set unchanged.
• The operations are: replacement (Rᵢ ↔ Rⱼ), interchange (Rᵢ ↔ Rⱼ), and scaling (cRᵢ → Rᵢ) where c is a nonzero constant.
• These operations can be used to simplify an augmented matrix to reveal the solution to the system

Example illustrating finding the solutions of a linear system

• A method of using elementary row operations to transform an augmented matrix into row echelon form, and further to reduced row echelon form allows determination of solutions.