Area and volume | Wild Linear Algebra A 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.

CONTENT SUMMARY: pg 1: @00:08 area and volume; setting up affine geometry as independent of distance;
pg 2: @02:03 area of a parallelogram;
pg 3: @06:35 general formula for area of a parallelogram; one of the most important formulas in mathematics;
pg 4: @11:53 algebraic approach to measuring area; bi-vector (Herman Grassmann); Grassmann algebra;
pg 5: @15:11 Bi-vectors; torque;
pg 6: @21:14 linear momentum; conservation of momentum; momentum and force; Bi-vectors; angular momentum; torque
pg 7: @25:52 Bi-vector and electromagnetism; cross_product mentioned;
pg 8: @28:42 Bi-vectors in the plane; operations on bi-vectors;
pg 9: @30:59 bi-vector Distributive laws:
pg 10: @34:04 claim: In the plane, every bi-vector is a multiple of the bi-vector of base vectors; geometric proof; algebraic proof
pg 11: @39:29 example using the result of the previous page; in affine geometry area is a relative concept;
pg 12: @42:44 2 more examples; Ratios of areas are affine geometry invariants;
pg 13: @45:52 3_dim affine space; vector, bi-vector, tri-vector; pg 14: @49:07 using tri-vectors to compute a volume;
pg 15: @53:24 exercises 4.(1:2);
pg 16: @54:43 exercise 4.3. (THANKS to EmptySpaceEnterprise)

Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.

Please consider supporting this Channel bringing you high quality mathematics lectures by becoming a Patron at https://www.patreon.com/njwildberger?

A screenshot PDF which includes WildLinAlg1 to 26 can be found at my WildEgg website here: http://www.wildegg.com/store/p107/product-screenshot-Wild-Linear-Algebra-A-pdf

My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/

Screenshot PDFs help you navigate through most of Norman’s video series: they can be found at http://www.wildegg.com/wildegg-store….

My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/…. I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?… .

Questions?