Mathematics of Systems of Linear Equations

Study Notes

Systems of linear equations: An arbitrary system of m linear equations in n unknowns can be written as a matrix equation: Ax = b.
A is the coefficient matrix, b is the constant matrix, and x is the unknown vector.
The augmented matrix [Ab] includes both the coefficients and constants. The number of rows in A equals the number of equations, and the number of columns equals the number of unknowns.

Methods: Two common methods for solving a system of linear equations using matrices are Gaussian elimination and Gauss-Jordan elimination.
Gaussian elimination: This method puts the augmented matrix into row-echelon form, making the solution set apparent through inspection.
Gauss-Jordan elimination: Similar to Gaussian elimination, but results in a reduced row-echelon form. The augmented matrix is manipulated further to have only leading 1's and 0's above and below them.
Elementary row operations: Used to transform the augmented matrix during both Gaussian and Gauss-Jordan elimination. These operations include swapping rows (R_i ↔ R_j), multiplying a row by a non-zero constant (kR_i→R_i), and adding one row multiplied by a constant to another row (R_i +kR_j→R_i).

Consistent vs. Inconsistent: A system of linear equations is consistent if it has at least one solution. Otherwise, it's inconsistent.
Unique Solution: A unique solution occurs when each variable has a specific value.
Infinitely Many Solutions: Infinitely many solutions occur when there are free variables. The system has more unknowns than equations.
Leading and Free Variables: In the augmented matrix, variables that lead to the presence of 1 in the solution matrix are called leading variables. Those that are not leading are referred to as free variables, and free variables are assigned to arbitrary values for a variety of solutions.
Parameters: Free variables are commonly represented with parameters, like t,s, etc.
Row-Echelon Form: Essential form for solving the system of equations. It has one or more leading 1's in each row.
General Solutions: Used to represent all possible solutions to a linear system when there are infinitely many solutions. This involves expressing the leading variables in terms of the free variables.