Linear Algebra - Linear Equation

> Linear Algebra

1 - About

A linear equation represnts a linear function that forms a straight line.

A common form of a linear equation in the two variables (two dimensions) x and y is

<math> y= mx + b </math>

where:

  • m is a constant named the slope or gradient of the line
  • b is a constant named the intercept. It determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.

In Statistics, it's the basic of a regression

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3 - Assumptions

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, y1/3, and sin(x)

4 - Linear Equation in

4.1 - Vector

In linear algebra, a linear equation can be expressed as an equation stating the value of the dot-product of:

  • a coefficient vector (a vector whose entries are the coefficients)
  • and a vector of unknowns variables.

Therefore a linear equation (function) is an equation of the form

<math>a . x = \beta</math>

where:

  • a is a coefficient vector (for instance <math> 1, 4, -3, \dots, 2</math>),
  • x is a vector of unknowns variables (for instance <math> x, y, z</math> or <math> x_1, \dots, x_n</math>)
  • and <math>\beta</math> is a scalar (for instance, 3)
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5 - Type

5.1 - Homogeneous

A linear equation a · x = 0 with zero right-hand side is a homogeneous linear equation. A system of homogeneous linear equations is called a homogeneous linear system.

6 - Functions

Oracle:

  • Slope: REGR_SLOPE(Y,X)
  • Intercept: REGR_INTERCEPT(Y,X)

7 - Documentation / Preference

linear_algebra/linear_equation.txt · Last modified: 2017/09/13 16:04 by gerardnico