# Linear Algebra - Linear Equation

A linear equation represents 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

$y= mx + b$

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

## 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)

## Linear Equation in

### 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

$a . x = \beta$

where:

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

## Type

### 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.

## Functions

Oracle:

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