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)
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