Linear function

As a polynomial function

Main article: Linear function (calculus)

In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial. (The latter is a polynomial with no terms, and it is not considered to have degree zero.)

When the function is of only one variable, it is of the form :f(x)=ax+b, where a and b are constants, often real numbers. The graph of such a function of one variable is a nonvertical line. a is frequently referred to as the slope of the line, and b as the intercept.

If a 0 then the gradient is positive and the graph slopes upwards.

If *a

For a function f(x_1, \ldots, x_k) of any finite number of variables, the general formula is :f(x_1, \ldots, x_k) = b + a_1 x_1 + \cdots + a_k x_k , and the graph is a hyperplane of dimension k.

A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one variable, is a horizontal line.

In this context, a function that is also a linear map (the other meaning of linear functions, see the below) may be referred to as a homogeneous linear function or a linear form. In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valued affine maps.

As a linear map

Main article: Linear map

In linear algebra, a linear function is a map f from a vector space \mathbf{V} to a vector space \mathbf{W} (Both spaces are not necessarily different.) over a same field K such that :f(\mathbf{x} + \mathbf{y}) = f(\mathbf{x}) + f(\mathbf{y}) :f(a\mathbf{x}) = af(\mathbf{x}). Here a denotes a constant belonging to the field K of scalars (for example, the real numbers), and x and y are elements of \mathbf{V}, which might be K itself. Even if the same symbol + is used, the operation of addition between x and y (belonging to \mathbf{V}) is not necessarily same to the operation of addition between f\left( \mathbf{x} \right) and f\left( \mathbf{y} \right) (belonging to \mathbf{W}).

In other terms the linear function preserves vector addition and scalar multiplication.

Some authors use "linear function" only for linear maps that take values in the scalar field; these are more commonly called linear forms.

The "linear functions" of calculus qualify as "linear maps" when (and only when) , or, equivalently, when the constant b equals zero in the one-degree polynomial above. Geometrically, the graph of the function must pass through the origin.

Notes

References

• Izrail Moiseevich Gelfand (1961), Lectures on Linear Algebra, Interscience Publishers, Inc., New York. Reprinted by Dover, 1989.
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• Leonid N. Vaserstein (2006), "Linear Programming", in Leslie Hogben, ed., Handbook of Linear Algebra, Discrete Mathematics and Its Applications, Chapman and Hall/CRC, chap. 50.

References

1. "The term ''linear function'' means a linear form in some textbooks and an affine function in others." Vaserstein 2006, p. 50-1
2. Stewart 2012, p. 23
3. A. Kurosh. (1975). "Higher Algebra". Mir Publishers.
4. T. M. Apostol. (1981). "Mathematical Analysis". Addison-Wesley.
5. Shores 2007, p. 71
6. Gelfand 1961