Explore the fundamental concepts of area and volume in linear algebra through this comprehensive lecture. Delve into the pictorial treatment of area before transitioning to an algebraic formulation using bi-vectors, introduced by Grassmann in the 1840s. Discover the central role these concepts play in underpinning the entire subject and their connection to the rich theory of determinants. Learn about the general formula for the area of a parallelogram, considered one of the most important formulas in mathematics. Investigate bi-vectors and their applications in physics, including torque, linear momentum, and electromagnetism. Examine the properties and operations of bi-vectors in the plane, including distributive laws and their relationship to base vectors. Extend the concepts to three-dimensional space, exploring tri-vectors and their use in computing volume. Gain practical experience through exercises that reinforce the learned material, providing a solid foundation for further study in linear algebra and related fields.
Overview
Syllabus
CONTENT SUMMARY: pg 1: @ area and volume; setting up affine geometry as independent of distance;
pg 2: @ area of a parallelogram;
pg 3: @ general formula for area of a parallelogram; one of the most important formulas in mathematics;
pg 4: @ algebraic approach to measuring area; bi-vector Herman Grassmann; Grassmann algebra;
pg 5: @ Bi-vectors; torque;
pg 6: @ linear momentum; conservation of momentum; momentum and force; Bi-vectors; angular momentum; torque
pg 7: @ Bi-vector and electromagnetism; cross_product mentioned;
pg 8: @ Bi-vectors in the plane; operations on bi-vectors;
pg 9: @ bi-vector Distributive laws:
pg 10: @ claim: In the plane, every bi-vector is a multiple of the bi-vector of base vectors; geometric proof; algebraic proof
pg 11: @ example using the result of the previous page; in affine geometry area is a relative concept;
pg 12: @ 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: @ using tri-vectors to compute a volume;
pg 15: @ exercises 4.1:2;
pg 16: @ exercise 4.3. THANKS to EmptySpaceEnterprise
Taught by
Insights into Mathematics
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