Hi all, This Blog is an English archive of my PhD experience in Imperial College London, mainly logging my research and working process, as well as some visual records.

Monday 3 September 2007

Linear function

In mathematics, the term linear function can refer to either of two different but related concepts.

Usage in elementary mathematics

In elementary algebra and analytic geometry, the term linear function is often used to mean a first degree polynomial function of one variable. These functions are called "linear" because they are precisely the functions whose graph in the Cartesian coordinate plane is a straight line.

Such a function can be written as

f(x) = mx + b,

where m and b are real constants and x is a real variable. The constant m is often called the slope while b is the y-intercept, which gives the point of intersection between the graph of the function and the y-axis. Changing m makes the line steeper or shallower, while changing b moves the line up or down.

Three geometric linear functions — the red and blue ones have same slope (m), while red and green ones have same y-intercept (b).

Three geometric linear functions — the red and blue ones have same slope (m), while red and green ones have same y-intercept (b).

Examples of functions whose graph is a line include the following:

  • f1(x) = 2x + 1
  • f2(x) = x / 2 + 1
  • f3(x) = x / 2 − 1

The graphs of these are shown in the image at right.

Usage in advanced mathematics

In advanced mathematics, a linear function often means a function that is a linear map, that is, a map between two vector spaces that preserves vector addition and scalar multiplication.

A function f(x) = mx + b is a linear map if and only if b = 0. For other values of b this falls in the more general class of affine maps.

1 comment:

rajput said...

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