Inmathematics, the termlinear function refers to two distinct but related notions:^[1]

• Incalculus and related areas, alinear function is afunction whosegraph is astraight line, that is, apolynomial function ofdegree zero (a constant polynomial) or one (a linear polynomial).^[2] For distinguishing such a linear function from the other concept, the termaffine function is often used.^[3]
• Inlinear algebra,mathematical analysis,^[4] andfunctional analysis, alinear function is a kind of function betweenvector spaces.^[5]

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

When the function is of only onevariable, it is of the form

${\displaystyle f(x)=ax+b,}$

wherea andb areconstants, oftenreal numbers. Thegraph of such a function of one variable is a nonvertical line.a is frequently referred to as the slope of the line, andb as the intercept.

Ifa > 0 then thegradient is positive and the graph slopes upwards.

Ifa < 0 then thegradient is negative and the graph slopes downwards.

For a function${\displaystyle f(x_1,\dots ,x_k)}$ of any finite number of variables, the general formula is

${\displaystyle f(x_1,\dots ,x_k)=b+a_1{x}_1+\cdots +a_k{x}_k,}$

and the graph is ahyperplane of dimensionk.

Aconstant 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 ahomogeneous linear function or alinear form. In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valuedaffine maps.

In linear algebra, a linear function is a map${\displaystyle f}$ from avector space${\displaystyle \mathbf{V}}$ to a vector space${\displaystyle \mathbf{W}}$ (Both spaces are not necessarily different.) over a samefieldK such that

${\displaystyle f(\mathbf{x}+\mathbf{y})=f(\mathbf{x})+f(\mathbf{y})}$

${\displaystyle f(a\mathbf{x})=af(\mathbf{x}).}$

Herea denotes a constant belonging to the fieldK ofscalars (for example, thereal numbers), andx andy are elements of${\displaystyle \mathbf{V}}$, which might beK itself. Even if the same symbol${\displaystyle +}$ is used, the operation of addition betweenx andy (belonging to${\displaystyle \mathbf{V}}$) is not necessarily same to the operation of addition between${\displaystyle f\left(\mathbf{x}\right)}$ and${\displaystyle f\left(\mathbf{y}\right)}$ (belonging to${\displaystyle \mathbf{W}}$).

In other terms the linear function preservesvector addition andscalar multiplication.

Some authors use "linear function" only for linear maps that take values in the scalar field;^[6] these are more commonly calledlinear forms.

The "linear functions" of calculus qualify as "linear maps" when (and only when)f(0, ..., 0) = 0, or, equivalently, when the constantb equals zero in the one-degree polynomial above. Geometrically, the graph of the function must pass through the origin.

• Homogeneous function
• Nonlinear system
• Piecewise linear function
• Linear approximation
• Linear interpolation
• Discontinuous linear map
• Linear least squares

• Izrail Moiseevich Gelfand (1961),Lectures on Linear Algebra, Interscience Publishers, Inc., New York. Reprinted by Dover, 1989.ISBN 0-486-66082-6
• Shores, Thomas S. (2007).Applied Linear Algebra and Matrix Analysis.Undergraduate Texts in Mathematics. Springer.ISBN 978-0-387-33195-9.
• Stewart, James (2012).Calculus: Early Transcendentals (7E ed.). Brooks/Cole.ISBN 978-0-538-49790-9.
• Leonid N. Vaserstein (2006), "Linear Programming", inLeslie Hogben, ed.,Handbook of Linear Algebra, Discrete Mathematics and Its Applications, Chapman and Hall/CRC, chap. 50.ISBN 1-584-88510-6

• Polynomial functions

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