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)