Linearity

In mathematics, the term linear is used in two distinct senses for two different properties:

<ul><li>linearity of a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> (or <a href="/facts/Mapping_(mathematics)/FABF1l2f">mapping</a>);</li>
<li>linearity of a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a>.</li></ul>
An example of a linear function is the function defined by 
 
 
 
 f
 (
 x
 )
 =
 (
 a
 x
 ,
 b
 x
 )
 
 
 {\displaystyle f(x)=(ax,bx)}
 
 that maps the real line to a line in the <a href="/facts/Euclidean_plane/tAUtRFcn">Euclidean plane</a> R2 that passes through the origin. An example of a linear polynomial in the variables 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 and 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
 is 
 
 
 
 a
 X
 +
 b
 Y
 +
 c
 Z
 +
 d
 .
 
 
 {\displaystyle aX+bY+cZ+d.}
 
 
Linearity of a mapping is closely related to <a href="/facts/Proportionality_(mathematics)/xDQmfh6t">proportionality</a>. Examples in <a href="/facts/Physics/a7aH2y08">physics</a> include the linear relationship of <a href="/facts/Voltage/EN2po6N2">voltage</a> and <a href="/facts/Electric_current/Dc1xdnUX">current</a> in an <a href="/facts/Electrical_conductor/BSuyupLP">electrical conductor</a> (<a href="/facts/Ohm%2527s_law/PjfEyTgI">Ohm's law</a>), and the relationship of <a href="/facts/Mass/HHdc8XmF">mass</a> and <a href="/facts/Weight/9NMHCsQd">weight</a>. By contrast, more complicated relationships, such as between <a href="/facts/Velocity/Q3o1LVDe">velocity</a> and <a href="/facts/Kinetic_energy/aJRxUd4p">kinetic energy</a>, are <a href="/facts/Nonlinear_system/4aYItALC">nonlinear</a>.
Generalized for functions in more than one <a href="/facts/Dimension_(mathematics)/0ktmUyac">dimension</a>, linearity means the property of a function of being compatible with <a href="/facts/Addition/apFWJBFB">addition</a> and <a href="/facts/Scale_analysis_(mathematics)/xwq3Mxej">scaling</a>, also known as the <a href="/facts/Superposition_principle/GKGs9C92">superposition principle</a>.
Linearity of a polynomial means that its <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> is less than two. The use of the term for polynomials stems from the fact that the <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a> of a polynomial in one variable is a straight <a href="/facts/Line_(geometry)/gJqsr4Gj">line</a>. In the term "<a href="/facts/Linear_equation/JfCgALwf">linear equation</a>", the word refers to the linearity of the polynomials involved.
Because a function such as 
 
 
 
 f
 (
 x
 )
 =
 a
 x
 +
 b
 
 
 {\displaystyle f(x)=ax+b}
 
 is defined by a linear polynomial in its argument, it is sometimes also referred to as being a "linear function", and the relationship between the argument and the function value may be referred to as a "linear relationship". This is potentially confusing, but usually the intended meaning will be clear from the context.
The word linear comes from <a href="/facts/Latin/pM3Kge85">Latin</a> linearis, "pertaining to or resembling a line".

Linearity open-in-new

Linearity