Linear function

In mathematics, a linear function is a special type of function that is used in many areas of math and science. It can mean two slightly different things, depending on what you are studying.

One meaning comes from calculus and related subjects. Here, a linear function is a function whose graph makes a straight line. This kind of function can be described with a simple math rule. You multiply inputs by certain numbers and then add them together. These functions are part of a group called polynomial functions.

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 work together. Both meanings of linear functions are important for solving real-world problems and understanding patterns.

As a polynomial function

In calculus, a linear function is a type of polynomial with a degree of one or less. It looks like the equation 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 shows how steep the line is, called the slope. The number b shows 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.