Analogous to a linear form, a bilinear form is a Bilinear map where the image is the underlying field of the vector space, e.g. .
Some definitions require both of the input spaces to be the same, e.g. , but it doesn't make much different in general.
The most important example of a bilinear form is the dot product. It is only defined if both the input spaces are the same.
Film about the financial crisis of 2007-2008 by 
Ciro Santilli 35 Updated 2025-01-29 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-29 +Created 1970-01-01The Men Who Stole the World by Benoît Bringer
. This timestamp contains a good explanation of how banks were knowingly reselling bad loans at incredible profits. It features Richard Bowen, previously from Citi Group compliance. System of linear ordinary differential equations by Ciro Santilli 35 Updated 2025-01-29 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-29 +Created 1970-01-01Ambush from ten sides performed by Jiaju Shen (2017)
Source. Ambush from ten sides performed live by Liu Dehai
. Source. The sound quality is not great, but you do get to see the master. TODO location and date. The Hundred Greatest Theorems by Paul and Jack Abad (1999) by Ciro Santilli 35 Updated 2025-01-29 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-29 +Created 1970-01-01With Telescopes however, it is possible. www.quora.com/Can-we-distinguish-individual-stars-in-other-galaxies-or-would-it-be-equivalent-to-say-we-know-there-are-other-forests-of-stars-galaxies-but-we-cant-tell-the-individual-trees-stars-What-is-the-farthest-individual/answer/Jerzy-Micha%C5%82-Pawlak contains an amazing answer that mentions two special cases of the furthest ones:
- gravitational lensing observation
- a star that is far but visible because its light is reflected by a nearby nebulae
But what we can definitely see are globular clusters of galaxies. E.g. the article en.wikipedia.org/wiki/Messier_87 basically gauges the size of galaxies by the number of globular clusters that they contain.
Nuclear physics is basically just the study of the complex outcomes of weak interaction + quantum chromodynamics.
Pinned article: ourbigbook/introduction-to-the-ourbigbook-project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivative - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. - Internal cross file references done right:
- Infinitely deep tables of contents:
Figure 6. Dynamic article tree with infinitely deep table of contents.Live URL: ourbigbook.com/cirosantilli/chordateDescendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact