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source: njwildberger 2011年3月7日

This is the full first lecture of a course on Linear Algebra. Given by N J Wildberger of the School of Mathematics and Statistics at UNSW, the course gives a more geometric and natural approach to this important subject, with lots of interesting applications. Our orientation is that Linear Algebra is really ``Linear Algebraic Geometry'': so teaching the algebra without the geometry is depriving the student of the heart of the subject.

The first lecture discusses the affine grid plane and introduces vectors, along with the number one problem of linear algebra: how to invert a linear change of coordinates!

Intended audience: first year college or undergraduate students, motivated high school students, high school teachers, general public interested in mathematics. Enjoy!

CONTENT SUMMARY: pg 1: @00:10 N. J. Wildberger Webpages:

web.maths.unsw.edu.au/~norman/index.html

pg 2: @02:10 Linear Algebra to Linear Algebraic Geometry; example; applications

pg 3: @04:52 2-dim affine geometry (no notion of perpendicularity); affine grid plane; the core problem of linear algebra

pg 4: @09:27 no distance measurement, no special point (origin), no angle

measurement; relative positions can be described; affine grid plane; vector;

connection between algebra and geometry

pg 5: @13:52 refinement of the affine grid plane; rational number

pg 6: @17:34 Multiples of a vector; adding vectors; basis vectors e1 and e2

pg 7: @22:34 Two affine grids and 2 sets of basis vectors

pg 8: @27:13 change of basis vectors example

pg 9: @30:47 change of basis example continued; Main Problem of Linear Algebra (MPLA)

pg 10: @34:03 summary of previous example; (MPLA) General case in 1dim; (MPLA) General case in 2dim

pg 11: @40:32 questions; Exercise 1.1

pg 12: @41:42 Exercises 1.2-4. (THANKS to EmptySpaceEnterprise)

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?... .

**WildLinAlg: A geometric course in Linear Algebra**

This course by N J Wildberger presents a geometrical view to Linear Algebra, with a focus on applications. We look at vectors, matrices, determinants, change of bases, row reduction, lines and planes, polynomial spaces, bases and coordinate vectors, and much more!

We are aiming for a careful exposition with an orientation on conceptual understanding, applications, and explicit examples. There are quite a few problems to challenge the viewer. It is perfectly possible to learn Linear Algebra from scratch with this course. I suggest to go through the series at about one a week, keep notes, and do all the problems.

1: Introduction to Linear Algebra 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

27: Geometry with linear algebra 28:12

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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