yt:playlist:PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2016-08-04T19:58:59+00:00 yt:video:fNk_zzaMoSs fNk_zzaMoSs UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2016-08-06T00:05:49+00:00 2025-06-09T19:09:13+00:00 Vectors | Chapter 1, Essence of linear algebra Beginning the linear algebra series with the basics. Help fund future projects: https://www.patreon.com/3blue1brown Music: https://vincerubinetti.bandcamp.com/track/grants-etude Thanks to Elo Marie Viennot and Ambros Gleixner from HTW Berlin (www.htw-berlin.de) for contributing German translations and dubbing. Thanks also to these viewers for their contributions to the subtitles Arabic: @Cewkins, Hazem Bengali: Md Rasheduzzaman Czech: @Chnapak Greek: Nikolaos Tsagkas Hebrew: Omer Tuchfeld Italian: mattiacolucci Spanish: Juan Carlos Largo Thai: own doggoV●ᴥ●V ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown yt:video:k7RM-ot2NWY k7RM-ot2NWY UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2016-08-06T23:49:28+00:00 2025-09-20T23:45:54+00:00 Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra The fundamental concepts of span, linear combinations, linear dependence, and bases. Help fund future projects: https://www.patreon.com/3blue1brown Thanks to Elo Marie Viennot and Ambros Gleixner from HTW Berlin (www.htw-berlin.de) for contributing German translations and dubbing. Thanks to these viewers for their contributions to translations Arabic: @Cewkins, Hazem Hebrew: Omer Tuchfeld Spanish: Juan Carlos Largo Vietnamese: @ngvutuan2811 ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown yt:video:kYB8IZa5AuE kYB8IZa5AuE UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2016-08-07T21:39:19+00:00 2025-09-05T06:31:44+00:00 Linear transformations and matrices | Chapter 3, Essence of linear algebra Quite possibly the most important idea for understanding linear algebra. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Home page: https://www.3blue1brown.com Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld Spanish: Juan Carlos Largo Vietnamese: @ngvutuan2811 ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown yt:video:XkY2DOUCWMU XkY2DOUCWMU UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2016-08-08T22:20:43+00:00 2025-09-09T07:04:16+00:00 Matrix multiplication as composition | Chapter 4, Essence of linear algebra Multiplying two matrices represents applying one transformation after another. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Home page: https://www.3blue1brown.com/ Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld Spanish: Juan Carlos Largo Vietnamese: @ngvutuan2811 ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown yt:video:rHLEWRxRGiM rHLEWRxRGiM UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2016-08-09T22:01:39+00:00 2025-09-22T06:25:07+00:00 Three-dimensional linear transformations | Chapter 5, Essence of linear algebra What do 3d linear transformations look like? Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Home page: https://www.3blue1brown.com/ Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown yt:video:Ip3X9LOh2dk Ip3X9LOh2dk UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2016-08-10T22:31:18+00:00 2025-09-22T06:31:39+00:00 The determinant | Chapter 6, Essence of linear algebra The determinant measures how much volumes change during a transformation. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Home page: https://www.3blue1brown.com/ Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld Vietnamese: @ngvutuan2811 ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown yt:video:uQhTuRlWMxw uQhTuRlWMxw UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2016-08-15T22:00:22+00:00 2025-09-22T07:05:20+00:00 Inverse matrices, column space and null space | Chapter 7, Essence of linear algebra How to think about linear systems of equations geometrically. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Home page: https://www.3blue1brown.com/ Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown yt:video:v8VSDg_WQlA v8VSDg_WQlA UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2016-08-16T21:59:23+00:00 2025-09-22T07:22:20+00:00 Nonsquare matrices as transformations between dimensions | Chapter 8, Essence of linear algebra A brief footnote on the geometric interpretation of non-square matrices. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Home page: https://www.3blue1brown.com/ Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld Vietnamese: @ngvutuan2811 ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown yt:video:LyGKycYT2v0 LyGKycYT2v0 UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2016-08-24T19:20:58+00:00 2025-09-16T07:58:42+00:00 Dot products and duality | Chapter 9, Essence of linear algebra Why the formula for dot products matches their geometric intuition. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Home page: https://www.3blue1brown.com/ Dot products are a nice geometric tool for understanding projection. But now that we know about linear transformations, we can get a deeper feel for what's going on with the dot product, and the connection between its numerical computation and its geometric interpretation. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld Vietnamese: @ngvutuan2811 ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown yt:video:eu6i7WJeinw eu6i7WJeinw UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2016-09-01T03:54:15+00:00 2025-10-06T07:43:59+00:00 Cross products | Chapter 10, Essence of linear algebra This covers the main geometric intuition behind the 2d and 3d cross products. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Home page: https://www.3blue1brown.com/ *Note, in all the computations here, I list the coordinates of the vectors as columns of a matrix, but many textbooks put them in the rows of a matrix instead. It makes no difference for the result, since the determinant is unchanged after a transpose, but given how I've framed most of this series I think it is more intuitive to go with a column-centric approach. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld Vietnamese: @ngvutuan2811 ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown yt:video:BaM7OCEm3G0 BaM7OCEm3G0 UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2016-09-01T03:54:15+00:00 2025-09-18T07:26:31+00:00 Cross products in the light of linear transformations | Chapter 11, Essence of linear algebra Why the formula for cross products matches the geometric intuition. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Home page: https://www.3blue1brown.com/ For anyone who wants to understand the cross-product more deeply, this video shows how it relates to a certain linear transformation via duality. This perspective gives a very elegant explanation of why the traditional computation of a dot product corresponds to its geometric interpretation. Minor error at 1:44, the third line of the matrix should read "v1 * w2 - w1 * v2" *Note, in all the computations here, I list the coordinates of the vectors as columns of a matrix, but many textbooks put them in the rows of a matrix instead. It makes no difference for the result since the determinant is unchanged after a transpose, but given how I've framed most of this series I think it is more intuitive to go with a column-centric approach. Full series: http://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld Vietnamese: @ngvutuan2811 ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown yt:video:jBsC34PxzoM jBsC34PxzoM UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2019-03-17T04:35:22+00:00 2025-09-13T06:11:54+00:00 Cramer's rule, explained geometrically | Chapter 12, Essence of linear algebra This rule seems random to many students, but it has a beautiful reason for being true. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Home page: https://www.3blue1brown.com/ Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld ---- If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. Music by Vincent Rubinetti. Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown yt:video:P2LTAUO1TdA P2LTAUO1TdA UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2016-09-11T17:56:20+00:00 2025-08-11T07:12:47+00:00 Change of basis | Chapter 13, Essence of linear algebra How do you translate back and forth between coordinate systems that use different basis vectors? Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Home page: https://www.3blue1brown.com/ Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support Thanks to these viewers for their contributions to translations Vietnamese: @ngvutuan2811 yt:video:PFDu9oVAE-g PFDu9oVAE-g UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2016-09-15T18:22:14+00:00 2025-06-09T03:12:59+00:00 Eigenvectors and eigenvalues | Chapter 14, Essence of linear algebra A visual understanding of eigenvectors, eigenvalues, and the usefulness of an eigenbasis. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Home page: https://www.3blue1brown.com Full series: https://3b1b.co/eola Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced. http://3b1b.co/support A solution to the puzzle at the end: https://www.dropbox.com/s/86yddvprfuaafju/Eigenvalue%20puzzle%20solution.pdf?dl=0 Typo: At 12:27, "more that a line full" should be "more than a line full". Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown yt:video:e50Bj7jn9IQ e50Bj7jn9IQ UCYO_jab_esuFRV4b17AJtAw 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw 2021-05-07T21:39:22+00:00 2025-07-30T14:19:04+00:00 A quick trick for computing eigenvalues | Chapter 15, Essence of linear algebra How to write the eigenvalues of a 2x2 matrix just by looking at it. Need a refresher on eigenvalues? https://youtu.be/PFDu9oVAE-g Thanks to Tim for the jingle: https://www.youtube.com/acapellascience Help fund future projects: https://www.patreon.com/3blue1brown​ An equally valuable form of support is to simply share the videos. Special thanks to these supporters: https://3b1b.co/quick-eigen-thanks Lockdown math lecture talking about the mean product formula: https://youtu.be/MHXO86wKeDY Timestamps: 0:00 - Background 4:53 - Examples 10:24 - Relation to the characteristic polynomial 12:00 - Last thoughts ------------------ These animations are largely made using a custom python library, manim. See the FAQ comments here: https://www.3blue1brown.com/faq#manim https://github.com/3b1b/manim https://github.com/ManimCommunity/manim/ You can find code for specific videos and projects here: https://github.com/3b1b/videos/ Music by Vincent Rubinetti. https://www.vincentrubinetti.com/ Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown