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)