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source: njwildberger 2014年3月17日
This is the first video of Part II of this course on linear algebra, and we give a brief overview of the applications which we will be concentrating on.
The first topic will be the connections between linear algebra and Euclidean and other geometries. Linear algebra provides an excellent framework for geometry, allowing Euclid's axiomatic approach to be replaced by logically more solid definitions and proofs. However for this to work seamlessly, a more algebraic approach than found in most texts will be here adopted. We will use ideas from Rational Trigonometry, and the dot product (or inner product) will play a central role.
We motivate these developments by going back to Euclid's understanding of mathematic's most important theorem: Pythagoras' theorem, and the intimate connection with the notion of perpendicularity.
27: Geometry with linear algebra 28:12
1: Introduction to Linear Algebra (N J Wildberger) 43:31
2: Geometry with vectors 44:15
3: Center of mass and barycentric coordinates 48:11
4: Area and volume 56:03
6: Applications of 2x2 matrices 43:47
5: Change of coordinates and determinants 48:36
7: More applications of 2x2 matrices 55:13
8: Inverting 3x3 matrices 45:44
9: Three dimensional affine geometry 43:02
10: Equations of lines and planes in 3D 1:08:52
11: Applications of 3x3 matrices 53:36
12: Generalized dilations and eigenvalues 55:35
13: Solving a system of linear equations 49:13
14: More row reduction with parameters 49:13
15: Applications of row reduction (Gaussian elimination) I 41:38
16: Applications of row reduction II 57:14
17: Rank and Nullity of a Linear Transformation 1:01:09
18: The geometry of a system of linear equations 1:08:59
19: Linear algebra with polynomials 46:14
20: Bases of polynomial spaces 59:50
21: More bases of polynomial spaces 45:52
22: Polynomials and sequence spaces 1:00:49
23: Stirling numbers and Pascal triangles 58:45
24: Cubic splines (Bezier curves) using linear algebra 32:35
25: Cubic splines using calculus 41:42
26: Change of basis and Taylor coefficient vectors 50:31
28: Dot products, Pythagoras' theorem, and generalizations 27:51
29: Applications of the dot product to planar geometry I 36:06
30: Applications of the dot product to planar geometry II 45:48
31: Circles and spheres via dot products I 41:58
32: Circles and spheres via dot products II 28:14
33: The relativistic dot product 35:46
34: Oriented circles and 3D relativistic geometry I 46:12
35: Oriented circles and relativistic geometry II 50:39
36: Energy, momentum and linear algebra 46:03
37: An elementary introduction to Special Relativity I 46:40
38: An elementary introduction to Special Relativity II 55:48
39: Length contraction, time dilation and velocity addition 1:00:40
40a: Relativistic dot products and complex numbers 38:06
40b: Relativistic dot products and complex numbers II 25:28
41: The chromatic algebra of 2x2 matrices I 32:14
42: The chromatic algebra of 2x2 matrices II 37:56
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