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In Plain Sight 1 Chapter 0 Preface Linear systems form the foundation of much of modern engineering and applied mathematics. This chapter introduces the basic concepts used throughout the text. Figure 1.1 — Graph of the linear function $y = mx + b$. I hear, I know; I see, I remember; I do, I understand. Definition A system is said to be linear if it satisfies additivity and homogeneity. Theorem 1.1 Every linear transformation between finite-dimensional vector spaces can be represented by a matrix. Example The system y = 3x is linear, whereas y = x² is not. ← Previous Chapter Chapter 1 Next Chapter →

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Figures and Equations Example Consider the linear relationship between two variables. The slope–intercept form is commonly written as $y = mx + b$. $$ y = mx + b $$ (1.1) Equation (1.1) describes a straight line with slope $m$ and intercept $b$. Figure 1.1 — Graph of the linear function $y = mx + b$. As shown in Figure 1.1, increasing the value of $m$ results in a steeper line. $$ \begin{aligned} Ax &= b \\ x &= A^{-1}b \end{aligned} $$ (1.2) Equation (1.2) represents the solution of a linear system when the matrix $A$ is invertible. y = m x + b

Table of Contents Table of Contents Chapter 1 — Introduction Chapter 2 — Vectors Chapter 3 — Linear Transformations A linear system is a mathematical model used to describe relationships between variables that obey the principle of superposition. 1 Definition A system is said to be linear if it satisfies additivity and homogeneity. Theorem 1.1 Every linear transformation between finite-dimensional vector spaces can be represented by a matrix. Example The system y = 3x is linear, whereas y = x² is not. This definition originates from classical linear algebra texts. ↩

Chapter 1 — Introduction Textbook Title Chapter 1 Chapter 1 Introduction to Linear Systems Linear systems form the foundation of much of modern engineering and applied mathematics. This chapter introduces the basic concepts used throughout the text. 1.1 Motivation Many real-world systems behave approximately linearly when examined over limited ranges. Simplicity enables understanding. ← Previous Page Chapter 1 Next Page →