Linear Algebra - (Line|Line Segment)

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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


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>


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


# 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>


To resume:


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.


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

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