Linear function

In mathematics, a linear function is a special type of function that has important uses in many areas of math and science. It can mean two slightly different things, depending on the context.

One meaning comes from calculus and related subjects. Here, a linear function is a function whose graph forms a straight line. This kind of function can be described using a simple math rule that involves multiplying inputs by certain numbers and then adding them together. These functions are part of a group called polynomial functions, and they are either constant (degree zero) or change at a steady rate (degree one). To avoid confusion with another meaning, some people use the term affine function.

In other areas like linear algebra, mathematical analysis, and functional analysis, a linear function describes a relationship between vector spaces. This helps mathematicians study complex spaces and how they interact in organized ways. Both meanings of linear functions are important tools for solving real-world problems and understanding patterns in many fields.

As a polynomial function

In calculus, a linear function is a type of polynomial with a degree of one or less. This includes simple equations of the form f(x) = ax + b, where a and b are fixed numbers. The graph of this function is a straight, non-vertical line. The number a tells us how steep the line is, called the slope, and b tells us where the line crosses the vertical axis, called the intercept.

In linear algebra, a linear function is a special kind of mapping between spaces called vector spaces. It follows two important rules: it preserves addition, meaning f(x + y) = f(x) + f(y), and it preserves multiplication by a constant, meaning f(ax) = af(x). These rules help us understand how the function behaves with vectors and scalars.