Essence of linear algebra preview

This introduces the "Essence of linear algebra" series, aimed at animating the geometric intuitions underlying many of the topics taught in a standard linear algebra course.
Error corrections:
- At one point I mistakenly allude to calculators using the Taylor expansion of sine for its computations, when in reality most use CORDIC (or something like it).
- Around 30 seconds in, there is a typo in how the determinant is written, which should be ad - bc

Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced.
Thanks to these viewers for their contributions to translations
French: @bapraAlbert
Hebrew: Omer Tuchfeld

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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).

Various social media stuffs:

[serock]

I just got my grade in linear algebra, highest score. Couldn't have done it without these videos, they didn't just make the subject much easier, but also more intuitive and fun.

[VincentRubinetti]

This video illustrates the big problem with how we teach math in the United States. Teachers and professors leap right into equations and numbers, completely skipping any conceptualization of them such as illustrations or high level explanations.

For example, I remember back to my algebra class in high school when we first learned about functions. The teacher started to explain how to graph functions, and how to write them using f(x) notation. We learned more advanced concepts such as factoring, completing the square, etc. Not once did the teacher say anything to the effect of "a function is a thing you put a number into and get a result out". They never explained how a graph is a visualization of how a specific function's result changes based on the input. They only described the mechanics of it, and "how to do it".

It's no wonder when many kids get to calculus that they have no idea what's going on, because they still don't actually understand in their head what a function or graph really is. The unfortunate consequence is that so many people now think they're just bad at math, when really it was just never properly explained to them on a conceptual level, and the ones who did well in high school math were simply the ones who could make the conceptual leaps on their own.

[calaphos]

What a coincidence! Looking for videos to procrastinate from studying for my linear algebra exam I found this. Looks like I can combine these things now

[jlpicard]

I had an awful linear algebra instructor when I was working through my BS in engineering so this subject has been somewhat of an Achilles heal for me for many years. I could always do the math, but the geometric interpretations you provide here has really helped me develop the intuition that I have been striving for. Thanks for putting together this excellent series!

[iwainste]

These videos highlight where innovation in education can take us. Well done!!

[KhalilEstell]

Thank you for taking on this topic. My friend who is brilliant at maths told me 3 years before I took linear algebra about how it work geometrically, and I thought the subject was amazing. Then when it came time to learn it in college, I got almost none of that. I felt that either I missed out or I was just not understanding it as well as he did. But this video series has given me hope to understand and see Linear Algebra like he does.

Thank you. I can't wait!

[kazimierz]

You hit the nail on the head in this intro. I went through uni studying electrical engineering. In 1st year we did linear algebra and all we focused on was how to do various calculations. It was easy to memorize and I did very well but it wasn't until 3rd/4th year where linear algebra was applied to various topics like control systems that I realized how little I actually understood the topic. I understood the HOW but not the WHY or WHAT. Sadly this was over 10 years ago and I wish I had access to such wonderful resources like your videos. Keep up the great work!

[BondJamesTheGamer]

Just a small nitpick at 2:12, calculators and computers do not use Taylor polynomials to calculate trigonometric functions, but rather employ a method of using pre-calculated tables and values, called CORDIC, to find the answer

[car-keys]

Great timing! I am just about to start linear algebra in 2 weeks.

[KarnKaul]

I am starting out as a game engineer, and everything you said is absolutely correct. Although my linear algebra is decent, I now realise that the Achilles Heel of my high-school-undergrad learning is the fact that I always chose only ONE interpretation to understand a concept: as soon as one clicked, I never bothered to comprehend the others, because in my head, I knew at least one way to approach it. This has come back to bite me, because now I need to do the opposite: figure out how to use those tools to solve real-world problems (rather than been given a tool and a made up problem). I know unit vectors, dot products, and parametric forms, yet it took me a really long time to figure out how to easily determine if a given ray intersects a given a plane, in an actual job interview. I realised it's because of all those holes I've left in my knowledge of mathematics. It's quite disheartening! Is it even possible to develop a solid foundation now? :(

[contrazzed]

3.5 years deep into a bachelor's in computer science and I haven't taken a linear algebra course since high school. I've already passed calc II, probability and stats, discrete mathematics, and some logic courses. This information has been taken for granted in countless situations and it has always been a fuzzy subject in mathematics for me. Honestly, the visualization of the graph skewing blew my mind. It never clicked like that for me before. And yet it seems like such a simple thing to cover. Thank you for this course. I should probably be paying you thousands of dollars for this information.

[fobenga]

This is the kind of video that I REALLY need in the moment. I started Linear Algebra in college this month. I'm graduating in mechanical engineering.

[jdms]

I appreciate you making these videos so much! I've always asked professors about the geometric interpretations of linear algebra, especially eigenvalues, and never once did I get even an attempt at an answer. You are a godsend, sir.

[crclayton]

I really look forward to this series! My linear algebra class was all pure math, with proofs and matrices operations, and had to frantically catch up in my engineering courses. Your sin example was spot on, and really articulated some of the things I'd been feeling for a while!

[samgdotson]

Okay the analogy to having an intuitive feel for trig functions blew my mind. I'm taking linear algebra right now, and the geometric intuition about certain concepts is absolutely being watered down. This series is incredible for that. Thank you!

[sarahszabo]

I'm terrifically excited for this. I'm reviewing linear algebra over the summer so I can tutor it next semester :D

Kind of reminds me of when I was learning determinants, and I asked my professor what exactly a determinant was and why we were doing this strange sequence of numerical operations. He was totally stunned and didn't know exactly XD.

[kaiiiichen]

THANK YOU FOR THIS! I love how you talk about having a visual pedagogy or geometric understanding will make this easier. I have a TA in my class who just assumes I already understand the geometric understanding, so I'm grateful for your videos

[PepePopo]

This channel is just amazing, it might not get as many views as a popular channel that shows "hot pranks kiss" crap. But it has definitely a higher quality, please keep making these amazing videos. They have helped me a lot and I'm looking forward to this new series of algebra videos!!
Amazing work!

[bandar]

My advice to young students to take the benefit of these videos to have a deep understanding about linear algebra. Long time ago when I was an undergrad student (i.e. 2003), not only we had no ample visualized materials, but we couldn't access the Internet with a reasonable speed (i.e. dial-up connection and not available at our homes). We could solely determine eigenvalues and eigenvectors with no understanding about their meaning whatsoever . The worst came to the worst is that English is not our native language yet English is mandatory at our colleges. We were good at math because the math calculations doesn't require a strong linguistic background however, every thing we were good at has been ceased to exist. The reason was the fact that we didn't have an intuitive explanation about a lot of math materials that helps our memory to remember math concepts for a long time. We've wasted a lot of time on carrying out calculations which is supposed to be the task of the computer and forgotten to remember the main picture. My advice to students is that this time being an undergrad student will ever come back again. Learn materials and focus on the big picture and keep some notes for each course.

[DardanUkaj]

Wow... god bless you. I'm in Linear Algebra right now and I'm a prospective math major and I feel the same frustrations that you outlined in this video. My final is in two weeks!!