Area and volume | Geometric Linear Algebra 4 | NJ Wildberger

Area and volume in Linear Algebra are central concepts that underpin the entire subject, and lead naturally to the rich theory of determinants, a key subject of 18th and 19th century mathematics.

This is the fourth lecture of a first course on Linear Algebra, given by N J Wildberger.

Here we start with a pictorial treatment of area, then move to an algebraic formulation using bi-vectors. These are two-dimensional versions of vectors introduced in the 1840's by Grassmann.

The three dimensional case of volume uses tri-vectors.

Video Contents:
0:00 Intro
0:07 Overview
1:05 Area of Fundamental Figures
2:03 Area of a Parallelogram
6:34 Generalizing to an Arbitrary Parallelogram
11:52 Bi-Vectors
15:10 Physical Examples of Bi-Vectors 1
21:12 Physical Examples of Bi-Vectors 2
25:51 Physical Examples of Bi-Vectors 3
28:41 Bi-Vectors in the Plane 1
30:58 Bi-Vectors in the Plane 2
34:03 Unit Bi-Vectors and Area
39:27 Using Bi-Vectors to Compute Area
42:43 Invariance of Ratios of Areas in Affine Geometry
45:51 Three Dimensional Affine Space
49:05 Using Tri-Vectors to Compute Volume
53:23 Exercises

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