Linear Geometry and Algebra

2020 James B. Wilson

Colorado State University

 

Affine: data which has concepts of points, lines, intersections, parallel

no infinity.

Projective: data which has concepts of points, lines, intersections, and location of infinity

no parallel.

CC-By Attribution: Ourladyshigh at English Wikipedia

Linear Information (Geometry)

  • Two points determine a line
  • Two lines intersect in at most one point

It is purely a study of relationships

  • Lines are equations, \(y=3x+5\)
  • Lines created by points \((a,b),(c,d)\) are defined by formulas \(y-b=\frac{b-d}{a-c}(x-a)\)
  • Intersections are solutions to equations\[\begin{aligned} y & =3x+5\\ y &=-2x-1\end{aligned}\]

Linear Data (Algebra)

Main Issues

  • Algebra needs coordinates (basis)
  • Algebra needs operations (*,+,-,0)