Linear systems are a central topic in mathematics and applied sciences. A linear system consists of multiple linear equations involving the same set of variables. Mathematically, you can express a system of $n$ linear equations in $n$ unknowns as a matrix equation: $A x = b$, where $A$ is an $n \times n$ matrix of coefficients, $x$ is a column vector of unknowns, and $b$ is a column vector of constants. The existence and uniqueness of solutions to such a system depend on the properties of the matrix $A$. If $A$ is invertible (i.e., has full rank and a nonzero determinant), then the system has a unique solution. If $A$ is singular (determinant zero), the system may have no solution or infinitely many solutions, depending on the consistency of the equations.

To solve a system of linear equations numerically in R, you can use the solve() function, as shown above. Here, the matrix A represents the coefficients, and b is the vector of constants. The function solve(A, b) computes the solution vector x such that A x = b. This approach relies on matrix factorization methods under the hood, such as LU decomposition, to efficiently find the solution. The output gives you the values of the unknowns that satisfy all equations simultaneously.

It is important to consider numerical stability when solving linear systems computationally. If the matrix A is close to singular or has a very small determinant, small errors in the data or rounding errors during computation can lead to large errors in the solution. Such systems are called ill-conditioned. In practice, you should be cautious when interpreting solutions to ill-conditioned systems, as the computed solution might not accurately reflect the true mathematical solution due to the limitations of floating-point arithmetic.