Linear function

<h2 id="as-a-polynomial-function">As a polynomial function</h2>
Main article: <a href="/facts/Linear_function_(calculus)/sbyEbb6x">Linear function (calculus)</a>

In calculus, <a href="/facts/Analytic_geometry/3qWP0kM0">analytic geometry</a> and related areas, a linear function is a polynomial of degree one or less, including the <a href="/facts/Zero_polynomial/Lzak8VVx">zero polynomial</a> (the latter not being considered to have degree zero).
When the function is of only one <a href="/facts/Variable_(mathematics)/VbQlrvKz">variable</a>, it is of the form

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

where a and b are <a href="/facts/Constant_(mathematics)/SxlxTWFT">constants</a>, often <a href="/facts/Real_number/R02gw5Pb">real numbers</a>. The <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> 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 <a href="/facts/Slope/a2L3k0j5">gradient</a> is positive and the graph slopes upwards.
If a < 0 then the <a href="/facts/Slope/a2L3k0j5">gradient</a> is negative and the graph slopes downwards.
For a function 
 
 
 
 f
 (
 
 x
 
 1
 
 
 ,
 …
 ,
 
 x
 
 k
 
 
 )
 
 
 {\displaystyle f(x_{1},\ldots ,x_{k})}
 
 of any finite number of variables, the general formula is

f
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            k
          
        
        )
        =
        b
        +
        
          a
          
            1
          
        
        
          x
          
            1
          
        
        +
        ⋯
        +
        
          a
          
            k
          
        
        
          x
          
            k
          
        
        ,
      
    
    {\displaystyle f(x_{1},\ldots ,x_{k})=b+a_{1}x_{1}+\cdots +a_{k}x_{k},}

and the graph is a <a href="/facts/Hyperplane/2s2ycYYd">hyperplane</a> of dimension k.
A <a href="/facts/Constant_function/BmPx6dn7">constant function</a> 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 <a href="/facts/Homogeneous_function/wF9XX7s5">homogeneous</a> linear function or a <a href="/facts/Linear_form/oGh7Asrz">linear form</a>. In the context of linear algebra, the polynomial functions of degree 0 or 1 are the scalar-valued <a href="/facts/Affine_map/BD7yDr10">affine maps</a>.

<h2 id="as-a-linear-map">As a linear map</h2>
Main article: <a href="/facts/Linear_map/Q1PBKNVV">Linear map</a>

In linear algebra, a linear function is a map f between two <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> such that

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

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

Here a denotes a constant belonging to some <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> K of <a href="/facts/Scalar_(mathematics)/MkDbA4Eo">scalars</a> (for example, the <a href="/facts/Real_number/R02gw5Pb">real numbers</a>) and x and y are elements of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a>, which might be K itself.
In other terms the linear function preserves <a href="/facts/Vector_addition/bCLrdksy">vector addition</a> and <a href="/facts/Scalar_multiplication/jroFDfbM">scalar multiplication</a>.
Some authors use "linear function" only for linear maps that take values in the scalar field;<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> these are more commonly called <a href="/facts/Linear_form/oGh7Asrz">linear forms</a>.
The "linear functions" of calculus qualify as "linear maps" when (and only when) f(0, ..., 0) = 0, 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.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Homogeneous_function/wF9XX7s5">Homogeneous function</a></li>
<li><a href="/facts/Nonlinear_system/4aYItALC">Nonlinear system</a></li>
<li><a href="/facts/Piecewise_linear_function/6bKlCbEa">Piecewise linear function</a></li>
<li><a href="/facts/Linear_approximation/SDBAneFx">Linear approximation</a></li>
<li><a href="/facts/Linear_interpolation/galkvRGy">Linear interpolation</a></li>
<li><a href="/facts/Discontinuous_linear_map/W1BSWvau">Discontinuous linear map</a></li>
<li><a href="/facts/Linear_least_squares/dKctFUCV">Linear least squares</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Izrail Moiseevich Gelfand (1961), Lectures on Linear Algebra, Interscience Publishers, Inc., New York. Reprinted by Dover, 1989. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-66082-6</li>
<li>Shores, Thomas S. (2007). Applied Linear Algebra and Matrix Analysis. <a href="/facts/Undergraduate_Texts_in_Mathematics/MTjjHqWD">Undergraduate Texts in Mathematics</a>. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-33195-9.</li>
<li>Stewart, James (2012). Calculus: Early Transcendentals (7E ed.). Brooks/Cole. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-538-49790-9.</li>
<li>Leonid N. Vaserstein (2006), "Linear Programming", in <a href="/facts/Leslie_Hogben/XgV8X4Fk">Leslie Hogben</a>, ed., Handbook of Linear Algebra, Discrete Mathematics and Its Applications, Chapman and Hall/CRC, chap. 50. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-584-88510-6</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"The term linear function means a linear form in some textbooks and an affine function in others." Vaserstein 2006, p. 50-1 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Stewart 2012, p. 23 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">A. Kurosh (1975). Higher Algebra. Mir Publishers. p. 214. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">T. M. Apostol (1981). Mathematical Analysis. Addison-Wesley. p. 345. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Shores 2007, p. 71 <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Gelfand 1961 <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Linear function open-in-new

Linear function