Summary of Linear Systems: Resolution

Keywords

How to define a linear system?
What are the steps to apply Cramer's Method?
How does the Scaling Method solve linear systems?
How are matrices used to represent linear systems?
What is the importance of determinants in solving systems using Cramer's Method?
When does a linear system have a unique solution, infinite solutions, or no solution?
What are the iterative methods to solve linear systems?

Apply elementary operations on the rows of the augmented matrix $(A|b)$ until it reaches the scaled form.
Solve the resulting triangular system through successive substitutions.

Linear Systems: Set of linear equations with the same set of variables. They can be consistent (have a solution) or inconsistent (have no solution).
Cramer's Method: Algebraic method that uses determinants to solve square linear systems (same number of equations and unknowns).
Scaling Method: Technique that transforms the system into a simpler equivalent until it reaches a triangular form, facilitating the identification of solutions.
Matrices: Rectangular representations of numbers or functions that can be used to describe linear systems compactly.
Determinants: Unique number associated with a square matrix that is fundamental in solving systems using Cramer's Method.

Matrix Representation: Fundamental to understand linear systems and apply resolution methods based on matrices and determinants.
Existence and Uniqueness of Solution: Based on the Rouché-Capelli Theorem, which relates the rank of the coefficient matrix and the rank of the augmented matrix.
Iterative Methods: Such as Gauss-Seidel and Jacobi, important for large systems where direct methods like Cramer and Scaling may be unfeasible.

Calculation of Determinants: Laplace's development, Sarrus' rule (for 3x3 matrices), and properties that facilitate the calculation of determinants of higher order.
Rouché-Capelli Conditions: A system is compatible if the rank of the coefficient matrix is equal to the rank of the augmented matrix; if these are equal to the number of unknowns, the system is uniquely compatible (a single solution).

Solving a System using Cramer's Method:
- Given a 3x3 system, calculate the determinants of the coefficient matrix and the modified matrices for each variable.
- Find the variables by dividing the modified determinants by the determinant of the coefficient matrix.
Application of the Scaling Method:
- Transform the original system into scaled form using elementary row operations.
- Solve the resulting triangular system by substitution, starting from the last equation to the first.
Use of the Rouché-Capelli Theorem:
- Calculate the ranks of the coefficient and augmented matrices of a system to determine its compatibility and if it is determined or undetermined.

Linear Systems: Sets of linear equations that can have different types of solutions: unique, infinite, or none.
Matrix Representation: Fundamental for a compact view of the system and application of algebraic methods.
Cramer's Method: Uses determinants to solve systems with a single solution, applicable only when the coefficient matrix is square and its determinant is not zero.
Scaling Method: Transforms the system into a simplified triangular form, facilitating the solution through successive substitutions.
Determinants: Crucial in Cramer's Method and to verify the existence of unique solutions in the system.
Rouché-Capelli Theorem: Provides criteria to determine the existence and quantity of solutions based on the ranks of the coefficient and augmented matrices.

A linear system can be consistently solved if it has the same number of independent equations and unknowns.
Determinant calculation is essential in Cramer's Method and in the analysis of unique solutions of square systems.
The Scaling Method is versatile and applicable to any system, being especially useful for non-square systems or when the determinant of the coefficient matrix is zero.
Knowledge of elementary operations in matrices is essential for the application of the Scaling Method.
Iterative techniques can be the key to solving large systems, where direct methods are computationally costly.