Definition
Linearity is a property of a relationship or system where the output is directly proportional to the input. A linear function must satisfy two conditions:
- Additivity:
- How to read: “The function f of the quantity x plus y equals f of x plus f of y.”
- Meaning: The response to a sum of inputs is the sum of the individual responses.
- Homogeneity:
- How to read: “The function f of c times x equals c times f of x.”
- Meaning: Scaling the input by a factor c scales the output by the same factor.
Why It Matters
Linearity is the simplest form of predictability. Most of our mental models and basic tools (arithmetic, simple project plans, budgets) assume linearity. While the real world is often non-linear, linearity serves as the essential baseline for modeling and the first-order approximation required for complex analysis.
Core Concepts
- Superposition: The principle that the response of a linear system to multiple inputs is the sum of the responses to each input individually.
- Proportionality: If you double the input, the output exactly doubles.
- Invariance: The relationship (the “slope”) remains constant regardless of the magnitude or position of the input.
- State Independence: A linear system does not have feedback loops or memory that change its sensitivity over time.