linear systems

A vector \( b \in \mathbb{R}^n \) is said to be a linear combination of the vectors \( v_1, v_2, \dots, v_k \in \mathbb{R}^n \) if and only if there exist scalars \( c_1, c_2, \dots, c_k \in \mathbb{R} \) such that:

\[
b = c_1 v_1 + c_2 v_2 + \dots + c_k v_k
\]

Systems of linear equations are collections of linear equations that share the same unknowns and must be satisfied simultaneously.

Linear systems in two variables are sets of two linear equations with two unknowns, used to determine their common solutions through algebraic and graphical methods.

Cramer’s Rule provides a method for solving systems of linear equations in unknowns, by using the determinant of the system’s coefficient matrix. This rule applies only when the coefficient matrix is square and its determinant is non-zero, ensuring that the system has a unique solution.

The Gauss method is a systematic procedure for solving a system of \( n \) linear equations with \( n \) unknowns by successively eliminating variables using elementary row operations.

The Rouché–Capelli theorem characterizes the solvability of a system of linear equations by comparing the rank of the coefficient matrix with that of the augmented matrix, determining whether solutions exist and how many.