Linear Equations

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Linear Equations

In this chapter, we introduce linear equations, laying the foundation for the core themes of Analysis I.

System of Linear Equations

Definition: Linear Equations

Row Reduction and Echelon Forms

Definition: Echelon Form

Theorem: Uniqueness of the Reduced Echelon Form

Definition: Pivot positions

Theorem: Existence and Uniqueness Theorem

Vector Equations

Definition: Algebraic Properties of \(\mathbb{R}^n\)

Definition: subset of \(\mathbb{R}^n\)

The Matrix Equation \(\mathbf{A}\mathbf{x} = \mathbf{b}\)

Definition: Matrix equation

Theorem: \(\mathbf{m}\mathbf{X}\)

Solution Sets of Linear System

Linear Independence

Introduction to Linear Transformations

The Matrix of a Linear Transformation

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