Linear function

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 < 0 then the gradient is negative and the graph slopes downwards.

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) 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

In linear algebra, a linear function is a map f between two vector spaces s.t.

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 some field $K$ of scalars (for example, the real numbers) and $x$ and $y$ are elements of a vector space, which might be $K$ itself.

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;^[1] these are more commonly called linear forms.

The "linear functions" of calculus qualify as "linear maps" when (and only when) $f (0, ..., 0) = 0$ , or, equivalently, when the above constant $b$ equals zero. Geometrically, the graph of the function must pass through the origin.

Notes

^ Gelfand 1961

References

Izrail Moiseevich Gelfand (1961), Lectures on Linear Algebra, Interscience Publishers, Inc., New York. Reprinted by Dover, 1989. ISBN 0-486-66082-6
Thomas S. Shores (2007), Applied Linear Algebra and Matrix Analysis, Undergraduate Texts in Mathematics, Springer. ISBN 0-387-33195-6
James Stewart (2012), Calculus: Early Transcendentals, edition 7E, Brooks/Cole. ISBN 978-0-538-49790-9
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. ISBN 1-584-88510-6

[1] Gelfand 1961

[1]

As a linear map

See also

Notes

References