Systems of Linear Equations Summary

Topic Overview

  • Systems of Linear Equations (SLE) covered in lecture.

Linear Equation Definition

  • A linear equation has variables of the first power; represented as (y = mx + c).
  • Graphs of linear equations yield straight lines.

System of Linear Equations

  • Collection of two or more linear equations.
  • Represents (m) equations with (n) unknowns ((x1, x2, …, x_n)).
  • Coefficients represented as ((a{11}, a{12},…, a{mn})) and constants as ((b1, b2,…, bm)).

Types of Solutions

  • Unique Solution: Exactly one solution.
  • No Solution: Inconsistent system; parallel lines.
  • Infinitely Many Solutions: Collinear equations; one equation is a multiple of another.

Matrix Representation

  • General form: (AX = B), where:
    • (A) = coefficient matrix
    • (X) = matrix of variables
    • (B) = column matrix of constants.
  • Augmented matrix combines (A) and (B).

Methods for Solving SLE

  • Direct Methods:
    1. Simultaneous equations
    2. Inverse of coefficient matrix
    3. Cramer’s rule
  • Row Operations:
    1. Gaussian elimination
    2. Gauss-Jordan elimination.

Cramer's Rule

  • Applicable for invertible square matrices.
  • Solution for (AX = B) uses determinants of modified matrices (A_i).
  • Compute (x = \frac{|A1|}{|A|}), (y = \frac{|A2|}{|A|}).

Important Notes

  • Homogeneous systems: All R.H.S are zero.
  • Non-homogeneous: At least one R.H.S is not zero.
  • For a system of equations to be consistent, at least one solution must exist.

Notable Points

  • The determinant (det(A)) is crucial for determining the existence and uniqueness of solutions.