Linear Function Overview

Definitions and Context

• Linear Function has two main interpretations in mathematics:
  1. Calculus Perspective: A function whose graph is a straight line, a polynomial function of degree zero or one (affine function).
  2. Linear Algebra Perspective: A linear map, preserving vector addition and scalar multiplication.

As a Polynomial Function

• In calculus and analytic geometry, a linear function refers to:
  • A polynomial of degree one or less, including the zero polynomial.
  • One Variable Formula: f(x) = ax + b
    • a: slope of the line
    • b: intercept
    • Positive a: graph slopes upward
    • Negative a: graph slopes downward
  • Multiple Variables: Graph is a hyperplane.
  • Constant Function: Considered linear, degree zero, graph is a horizontal line.

Linear Function as a Linear Map

• In linear algebra:
  • A linear function is a map f between two vector spaces:
    • Preserves vector addition and scalar multiplication.
    • f(a*x + b*y) = a*f(x) + b*f(y)
  • Only considered a linear map when f(0, ..., 0) = 0 and b = 0 in a one-degree polynomial.

Key Concepts

• Homogeneous Linear Function / Linear Form: Refers to functions that are linear maps.
• Affine Functions: In linear algebra, polynomial functions of degree 0 or 1 are scalar-valued affine maps.

Additional Topics

• Integral of a function as a linear map.
• Discussion on nonlinear systems, piecewise linear functions, linear interpolation, and linear least squares.

References

• Gelfand (1961), Shores (2007), Stewart (2012), Vaserstein (2006)

This summary captures essential points about linear functions, providing a concise study guide for understanding their role in calculus and linear algebra.