Last modified: December 22, 2024
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Linear Systems of Equations
Linear systems of equations can be represented in a matrix form, which enables the use of a variety of numerical methods for solving them.
Mathematical Formulation
A linear system can be written as a set of equations:
$$ \begin{eqnarray} A_{11}x_1 + A_{12}x_2 + \ldots + A_{1n}x_n &=& b_1 \\ A_{21}x_1 + A_{22}x_2 + \ldots + A_{2n}x_n &=& b_2 \\ \vdots \\ A_{n1}x_1 + A_{n2}x_2 + \ldots + A_{nn}x_n &=& b_n \end{eqnarray} $$
This system of equations can be rewritten in a compact matrix form as Ax = b, where:
$$ A = \begin{bmatrix} A_{11} & A_{12} & \ldots & A_{1n} \\ A_{21} & A_{22} & \ldots & A_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ A_{n1} & A_{n2} & \ldots & A_{nn} \end{bmatrix}, \quad x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}, \quad b = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix} $$
Criteria for a Unique Solution
A system of equations has a unique solution if and only if:
- The determinant of A is not zero, i.e., det(A) ≠0, implying A is non-singular.
- The matrix A is invertible.
- The columns of A are linearly independent.
- The rows of A are linearly independent.
Example
Consider the system of equations:
$$ \begin{eqnarray} 3x + 2y - z &=& 1 \\ 2x - 2y + 4z &=& -2 \\ -x + 0.5y - z &=& 0 \end{eqnarray} $$
This can be represented as a matrix equation: Ax = b, where A, x, and b are defined as follows:
$$ A = \begin{bmatrix} 3 & 2 & -1 \\ 2 & -2 & 4 \\ -1 & 0.5 & -1 \end{bmatrix}, \quad x = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad b = \begin{bmatrix} 1 \\ -2 \\ 0 \end{bmatrix} $$
The system can be solved using various matrix methods such as the Gaussian elimination method, which involves transforming the matrix A into its row echelon form and then performing back substitution to solve for x.
Advantages
- Matrix methods for solving systems of equations are systematic and can be easily implemented in computer programs.
- Matrix representation is compact and insightful, especially when dealing with systems of many variables.
Limitations
- If the matrix is singular or nearly singular (i.e., det(A) is zero or close to zero), the system may have no solution or infinitely many solutions.
- Large systems of equations may be computationally expensive to solve using matrix methods.