Gramian: How to take the Determinant of a Rectangular Matrix, and Anki - Rubaiyat Khondaker

We know that you can use the determinant to figure out the area of a parallelogram in 2D, or the volume of a 3D parallelogram (a parallelepiped). But how would you compute the area of a parallelogram in 5 dimensions? Or a 3D parallelepiped in 7 dimensions?
It turns out that there's a concept in Linear Algebra called the Gramian Matrix, the associated determinant of which can be used to compute the above quantities. In this talk, we'll derive this concept in an intuitive way, including a few animations made using 3B1B's manim library, and see how to use it to generalise the surface integral formula. By the end you should know how to integrate along a 2D surface in 4 dimensions!

In addition to the maths, we'll briefly discuss spaced repetition - an evidence-based study technique that I've been using Anki to implement. While commonly used just for recall of knowledge, we'll see how Anki can be used to develop and improve long-term understanding of concepts in maths.

Some familiarity with concepts of vectors and matrices will be helpful.

1:08 Gramian matrix and determinant
53:38 Interlude: Learning with Anki
1:08:12 Gramian and surface integrals

--------------------------------------------------------------------------------
Subscribe: https://www.youtube.com/channel/UCdswONpWjfKTbpkBUP4fOyA?sub_confirmation=1
Our website: https://archim.soc.srcf.net/
Our Facebook page: https://www.facebook.com/archimedeans/
Join our mailing list: https://lists.cam.ac.uk/mailman/listinfo/soc-archimedeans

Видео Gramian: How to take the Determinant of a Rectangular Matrix, and Anki - Rubaiyat Khondaker канала The Archimedeans