Systems of Linear Equations

The backbone of linear algebra—systems of equations allow us to explore how variables relate and interact. Whether solving real-world problems or simplifying data, these systems bring structure to complexity. 🧩

Learning Objectives

Learning objectives of the Systems of Linear Equations section.

• Lines
• Linear System
• Matrices
• Vectors

Summary Table

Summary of the Systems of Linear Equations section.

Concept	Description	Example
Linear Equation	An equation in the form \(a_1x_1 + a_2x_2 + \cdots + a_nx_n = b\)	\(2x_1 - x_2 + x_3 = 8\)
System of Equations	A set of linear equations with shared variables	\(\begin{aligned} a_1x_1 + b_1x_2 &= c_1 \\ a_2x_1 + b_2x_2 &= c_2 \end{aligned}\)
Coefficient Matrix	A matrix containing only the coefficients of a system’s variables	\(\begin{bmatrix} 2 & -1 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 4 \end{bmatrix}\)
Augmented Matrix	A matrix combining coefficients and constants from a system	\(\begin{bmatrix} 2 & -1 & 1 & 8 \\ 1 & 0 & 0 & 2 \\ 0 & 0 & 4 & 7 \end{bmatrix}\)
Matrix Notation	General form to represent a matrix \(A\) with \(m\) rows and \(n\) columns	\(A = \begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{bmatrix}\)
Vector	An ordered list of numbers representing a point or direction in space	\(\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}\)
Column Vector	A vertical vector derived from a matrix column	\(\mathbf{a}_1 = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}\)
Row Vector	A horizontal vector derived from a matrix row	\(\mathbf{a}_1^T = \begin{bmatrix} 2 & -1 & 1 \end{bmatrix}\)