Linear Function

A Linear Function is an algebraic function composed solely of simple multiplication operations

• Context:
  • It can be a fitted linear function, if it was produced by a linear regression algorithm.
  • It can be a probabilistic linear function, if it has an error component.
  • It can define the shortest path between two points.
  • It can be a Linear Predictive Estimator produces by a Linear Modeling Task (associated to a linear model with free model parameters).
  • It can be used to describe a Linear Equation, such as [math]\displaystyle{ A \mathbf{x} = \mathbf{b} }[/math].
  • It can range from being a Continuous Linear Function to being a Discontinuous Linear Function (such as a piecewise linear function).
• Example(s):
  • [math]\displaystyle{ f(x) = -1.3 \times x + 1.4 }[/math].
  • a Multidimensional Linear Function, such as [math]\displaystyle{ f(x_1, x_2) = 3 \times x_1 + -1.3 \times x_2 + 1.4 }[/math].
  • a Constant Function, such as [math]\displaystyle{ f(x)=1.4 }[/math]
  • a Regressed Linear Function.
  • …
• Counter-Example(s):
  • a Nonlinear Function, such as: a polynomial function, such as [math]\displaystyle{ y = 3x^2 + 1.4 }[/math].
  • a Logistic Function.
  • a Linear Model, because it has ungrounded model parameters.
  • a Line-Output Function.
• See: Linear Regression, Linear Classification, Linear Map.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Linear_map#Definition_and_first_consequences Retrieved:2015-1-30.
  • Let V and W be vector spaces over the same field K. A function f: V → W is said to be a linear map if for any two vectors x and y in V and any scalar α in K, …

    A linear map from V to K (with K viewed as a vector space over itself) is called a linear functional.

2011

• (Wikipedia, 2011) ⇒ http://en.wikipedia.org/wiki/Linear_equation
  • A 'linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. Linear equations can have one or more variables. Linear equations occur with great regularity in applied mathematics. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state.

    A common form of a linear equation in the two variables x and y is [math]\displaystyle{ y = mx + b,\, }[/math] where m and b designate constants. The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane. In this particular equation, the constant m determines the slope or gradient of that line, and the constant term "b" determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.

    Since terms of linear equations cannot contain products of distinct or equal variables, nor any power (other than 1) or other function of a variable, equations involving terms such as xy, x², y^1/3, and sin(x) are nonlinear.

    … A linear equation can involve more than two variables. The general linear equation in n variables is: [math]\displaystyle{ a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b. }[/math] In this form, a₁, a₂, …, an are the coefficients, x₁, x₂, …, x_n are the variables, and b is the constant. When dealing with three or fewer variables, it is common to replace x₁ with just x, x₂ with y, and x₃ with z, as appropriate. Such an equation will represent an (n–1)-dimensional hyperplane in n-dimensional Euclidean space (for example, a plane in 3-space).

    In vector notation, this can be expressed as: [math]\displaystyle{ \overrightarrow{n} \cdot \overrightarrow{x} = \overrightarrow{n} \cdot \overrightarrow{x_0} }[/math] where [math]\displaystyle{ \overrightarrow{n} }[/math] is a vector normal to the plane, [math]\displaystyle{ \overrightarrow{x} }[/math] are the coordinates of any point on the plane, and [math]\displaystyle{ \overrightarrow{x_0} }[/math] are the coordinates of the origin of the plane.

2009

• http://en.wiktionary.org/wiki/linear_function
  • (Mathematics) Any function whose Graph is a Straight Line.
  • (Mathematics) Any function of the Sum of two Variables whose Value is the same as the sum of the values of the same function of the two variables singly

• http://en.wiktionary.org/wiki/straight_line
  • (Geometry) A line with no Curvature; a line with constant Direction. The Path of shortest length between two points.

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