Linear Algebra - (Line|Line Segment)

Card Puncher Data Processing

About

Geometry - Line in linear algebra.

Line and line segment definition in Linear Algebra that goes:

A line has a dimension of one because only one coordinate is needed to specify a point on it

Line vs Line Segment

A line segment has a scalar <math>\alpha</math> that ranges over a finite set of real number whereas a line has a scalar that ranges over all real numbers.

The below definitions applied to a line and to a line segment by changing the scalar range.

Through the origin

One-dimensional

The line through the origin and the vector v (the point v) is the set of points where the scalar <math>\alpha</math> ranges over all real numbers:

<math>\{\alpha. v : \alpha \in \mathbb{R}, 0 <= \alpha <= 1\}</math>

where:

forms the line through the origin and the vector v (ie the point)

The points of a line that does contain the origin, form a vector space

Three-dimensional

# Line in three dimension
{[x, y, z] : a * [x, y, z] = 0, b * [x, y, z] = 0}

Not through the origin

Definition of a line that goes through the vector u and v (where u and v are not the origin)

Translation and Scalar multiplication Formulation

Vector Translation and vector scalar multiplication are used to defined set of points forming an line segment (because the scalar <math>\alpha</math> ranges not in <math>\mathbb{R}</math> and is then in a finite set) that not necessarily go through the origin.

You translate the line by adding a vector c [0.5, 1] to every point of the line segment from [0, 0] to [3, 2].

<math>\{[0.5, 1] + \alpha [3, 2] : 0 <= \alpha <= 1\}</math>

Vector Addition Scalar Multiplication

The result is a line through c instead of through origin.

Symmetric Formulation (Affine,Convex)

A new formulation which is symmetric with respect to endpoints (ie you have a coefficient on [3, 2] and on [0.5, 1]) is:

<math>\{\alpha [3.5, 3] + \beta [0.5, 1] : (\alpha, \beta) \in \mathbb{R}, (\alpha, \beta) >= 0, \alpha + \beta = 1\}</math>

You get it by using distributivity: <math>\alpha(u+v)=\alpha.u+\alpha.v</math>

alpha[3,2]+[0.5,1] = alpha([3.5,1]-[0.5,1])+[0.5,1]
= alpha[3.5,1]+(1-alpha)[0.5,1]
= alpha[3.5,1]+beta[0.5,1]

This formulation describes the line that goes through the point [3.5, 3] and [0.5, 1].

More generally, the line through the vector u and v consists of the set of a affine combinations of u and v:

<MATH>\{\alpha{u} + \beta{v}: (\alpha, \beta) \in \mathbb{R}, \alpha + \beta = 1\}</MATH>

Combination

To resume:

Convex

A convex combination represents a Finite Line Segment.

An expression with two vectors of the form:

<math>\{\alpha.u + \beta.v : (\alpha, \beta) \in \mathbb{R}, (\alpha, \beta) >= 0, \alpha + \beta = 1\}</math>

is called a convex combination of u and v.

The u-to-v line segment consists of the set of convex combinations of u and v.

Affine

Infinite line Segment

delim{lbrace}{alpha u + beta v : alpha in bbR, beta in bbR, alpha + beta = 1}{rbrace}

An expression of the form alpha u + beta where alpha + beta = 1 is called an a affine combination of u and v.

The line through u and v consists of the set of affine combinations of u and v





Discover More
Support Vector Geometry
Data Mining - Support Vector Machines (SVM) algorithm

A support vector machine is a Classification method. supervised algorithm used for: Classification and Regression (binary and multi-class problem) anomalie detection (one class problem) Supports:...
Line In 3d
Geometry - Object

in Geometry An object can be: a pre-defined shape or an arbitrary one defined with the help of a path. Notions such as: The length of a vector The angle between two vectors. are introduced...
Card Puncher Data Processing
Geometry - Shape (with transformation and notion)

in Geometry A shape is any list of line drawing commands, which strung together, describe a continuous line or closed shape that can be filled or stroked. Move, Line, Quadratic and Cubic...
Card Puncher Data Processing
Linear Algebra - Linear combination

A linear combination of the vectors is the following expression: where: The scalars are the coefficients Every vector in is a linear combination: Every linear combination is a vector in...
Closest Point X Axis
Linear Algebra - Closest Point on a Line

How to find the closest point on a line from a point ? How to find the vector on the line that best approximates the given vector b (the closest point on the line) Letb be a vector Leta be a...
Card Puncher Data Processing
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 where: m is a constant named the slope...
Vector Arrow Multiplication
Linear Algebra - Scalar (Multiplication|Product) - Scaling

Scalar Multiplication (Scaling) is the multiplication of a vector (for instance ) by a scalar (real number) (for instance ) to produce another vector (for instance ) Multiplying a vector v by a scalar...
Card Puncher Data Processing
Linear Algebra - Vector Space (set of vector)

A vector space is a subset of the set of function representing a geometric objectpassing through the origin. A vector space over a field F is any set V of vector : with the addition and scalar-multiplication...
Vector Addition Arrow
Linear Algebra - Vector Vector Operations

Vector Vector Operations: (also known as addition or substraction) (Scaling) Using operator overloading, you can compute this operations with the following syntax. Operation Syntax...
Linear Vs True Regression Function
Machine Learning - Linear (Regression|Model)

Linear regression is a regression method (ie mathematical technique for predicting numeric outcome) based on the resolution of linear equation. This is a classical statistical method dating back more...



Share this page:
Follow us:
Task Runner