Systems of Linear Equations

Quick Recap: Introduction to Linear Systems, Augmented Matrices, and Linear Combinations

Solve the System of Linear Equations in 2D Using an Augmented Matrix

For this lesson, we begin the process of understanding notations and matrix setup for solving systems of equations.

Solve the System of Linear Equations in 2D

$x + 5y = 7, \quad -2x - 7y = -5$

$x_1 + 5x_2 = 7, \quad -2x_1 - 7x_2 = -5$

$\begin{bmatrix} 1 & 5 \\ -2 & -7 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 7 \\ -5 \end{bmatrix} \Rightarrow \begin{bmatrix} 1 & 5 \\ -2 & -7 \end{bmatrix}x = \begin{bmatrix} 7 \\ -5 \end{bmatrix} \Rightarrow \begin{bmatrix} 1 & 5 \\ -2 & -7 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 7 \\ -5 \end{bmatrix} \iff Ax = b.$

Coefficient Matrix:

$A = \begin{bmatrix} 1 & 5 \\ -2 & -7 \end{bmatrix}$

Coordinate vectors or mapping vectors … Also “variable vector” and “constant vector”

$x = \vec{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}, \quad b \equiv \vec{b} = \begin{bmatrix} 7 \\ -5 \end{bmatrix}.$

Augmented Matrix:

$\Rightarrow \begin{bmatrix} 1 & 5 & 7 \\ -2 & -7 & -5 \end{bmatrix}$

Elementary Row Operations:

$\begin{bmatrix} 1 & 5 & 7 \\ -2 & -7 & -5 \end{bmatrix}$

$2R_1 + R_2 \Leftarrow 2 \begin{bmatrix} 1 & 5 & 7 \end{bmatrix} + \begin{bmatrix} -2 & -7 & -5 \end{bmatrix} \Rightarrow \begin{bmatrix} 0 & 3 & 9 \end{bmatrix}$

$\sim \begin{bmatrix} 1 & 5 & 7 \\ 0 & 3 & 9 \end{bmatrix} - \frac{5}{3}R_2 + R_1 \sim \begin{bmatrix} 1 & 0 & -\frac{8}{3} \\ 0 & 3 & 9 \end{bmatrix}$

$\sim \begin{bmatrix} 1 & 0 & -\frac{8}{3} \\ 0 & 1 & 3 \end{bmatrix} \Rightarrow \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} -\frac{8}{3} \\ 3 \end{bmatrix}.$

$\text{span} \left\{ \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right\} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} c_1 + \begin{pmatrix} 0 \\ 1 \end{pmatrix} c_2 \quad \text{[linear combination]}.$

A system of linear equations consists of multiple equations involving the same set of variables. These systems are a cornerstone of linear algebra and can be analyzed and solved using matrix methods.

General Form

A system of $m$ linear equations in $n$ variables can be written as:

$\begin{aligned} a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n &= b_1, \\ a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n &= b_2, \\ &\vdots \\ a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n &= b_m. \end{aligned}$

In matrix form, this is expressed as:

$A\mathbf{x} = \mathbf{b},$

where:

$A$ is the $m \times n$ coefficient matrix,
$\mathbf{x}$ is the $n \times 1$ vector of variables,
$\mathbf{b}$ is the $m \times 1$ vector of constants.

Example

Consider the system:

$\begin{aligned} 2x + y &= 5, \\ 4x - 3y &= 6. \end{aligned}$

This can be written in matrix form as:

$\begin{bmatrix} 2 & 1 \\ 4 & -3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 6 \end{bmatrix}.$

Solving the System

Linear algebra provides several methods to solve systems of equations:

Gaussian Elimination: Use row reduction to transform the augmented matrix into reduced row echelon form (RREF) and solve for the variables.
Matrix Inversion: If $A$ is a square matrix and invertible, the solution is:
$\mathbf{x} = A^{-1}\mathbf{b}.$
Determinants: For small square systems, Cramer’s Rule uses determinants to find solutions:
$x_i = \frac{\det(A_i)}{\det(A)},$
where $A_i$ is the matrix obtained by replacing the $i$ -th column of $A$ with $\mathbf{b}$ .

Key Concepts

Existence and Uniqueness:
- If $\det(A) \neq 0$ , the system has a unique solution.
- If $\det(A) = 0$ , the system may have no solution or infinitely many solutions.
Consistency:
- A consistent system has at least one solution.
- An inconsistent system has no solutions.
Geometric Interpretation: In two variables, each equation represents a line. Solutions correspond to points where these lines intersect.

Applications

Systems of linear equations appear in numerous fields, including:

Physics: Modeling forces, circuits, and equilibrium problems.
Economics: Analyzing market equilibrium and optimization problems.
Computer Science: Solving linear programming and network flow problems.

Understanding these methods and concepts is essential for applying linear algebra to real-world problems.