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  • 00:00:08,095 --> 00:00:11,635 Hey, everyone! So I'm pretty excited about the next sequence of videos that I'm doing. It'll be about

    線型代数ほど基本的な理論はない 教授や教科書が途方もない行列計算で その簡潔さを難しく見せているにも関わらず ――ジャン・デュドネ

  • It'll be about linear algebra, whichas a lot of you knowis one of those subjects that's required knowledge for

    こんにちは! このビデオシリーズを作れることにとても興奮しています

  • just about any technical discipline, but it's also—I've noticedgenerally poorly understood by

    これから線型代数について説明していきます ご存知のように線型代数は

  • students taking it for the first time. A student might go through a class and learn how to compute

    あらゆる分野で必要とされますが 初めて学ぶ生徒は

  • lots of things, like matrix multiplication, or the determinant, or cross productswhich use the

    あまりよく理解していません 授業では,多くの計算方法を学びます

  • determinantor eigenvalues, but they might come out without really understanding why matrix


  • multiplication is defined the way that it is, why the cross product has anything to do with the

    固有値などです しかし生徒は,なぜ行列のかけ算が

  • determinant, or what an eigenvalue really represents.

    このように定義されるのか 外積と行列式はどういう関係なのか

  • Often times, students end up well-practiced in the numerical operations of matrices, but are only


  • vaguely aware of the geometric intuitions underlying it all. But there's a fundamental difference


  • between understanding linear algebra on a numerical level and understanding it on a geometric level.


  • Each has its place, butroughly speakingthe geometric understanding is what lets you judge what

    しかし,「数値計算ができること」と 「幾何学的な理解ができること」は根本的にちがいます

  • tools to use to solve specific problems, feel why they work, and know how to interpret the results,

    それぞれ利点はありますが 幾何学的な理解により,問題を解くためにどの方法を使えばよいか判断できるようになります

  • and the numeric understanding is what lets you actually carry through the application of those tools.


  • Now, if you learn linear algebra without getting a solid foundation in that geometric understanding,


  • the problems can go unnoticed for a while, until you've gone deeper into whatever field you happen to


  • pursue, whether that's computer science, engineering, statistics, economics, or even math itself.


  • Once you're in a class, or a job for that matter, that assumes fluency with linear algebra, the way


  • that your professors or your co-workers apply that field could seem like utter magic.

    線型代数を使いこなせることが 要求される教室や職場では

  • They'll very quickly know what the right tool to use is, and what the answer roughly looks like,

    教授や同僚が線型代数を使いこなすのが 魔法のように見えるでしょう

  • in a way that would seem like computational wizardry if you assumed that they're actually

    彼らは,どの方法を使えばよいか 大まかな答えは何か,すぐ知ることができるため

  • crunching all the numbers in their head.


  • As an analogy, imagine that when you first learned about the sine function in trigonometry, you were


  • shown this infinite polynomial. This, by the way, is how your calculator evaluates the sine function.


  • For homework, you might be asked to practice computing approximations to the sine

    この無限級数を示されたとします (電卓はこれを使って三角関数を計算しています)

  • function, by plugging various numbers into the formula and cutting it off at a reasonable point.

    宿題としてサイン関数のおおまかな値を 計算してくるように言われます

  • And, in fairness, let's say you had a vague idea that this was supposed to be related to triangles,

    さまざまな値を公式に代入し 適当な項まで計算するのです

  • but exactly how had never really been clear, and was just not the focus of the course. Later on, if

    そして,これが三角形に関係しているかもしれないと 気づいたとしましょう

  • you took a physics course, where sines and cosines are thrown around left and right, and people are

    しかし正確にはっきりとは理解できず しかもこの授業のポイントでもありません

  • able to tell pretty immediately how to apply them, and roughly what the sine of a certain value is,


  • it would be pretty intimidating, wouldn't it? It would make it seem like the only people who are cut

    すぐにその使い方を知ることができます サインがどのくらいの値になるかもわかります

  • out for physics are those with computers for brains, and you would feel unduly slow or dumb for

    こわいことではないですか? まるで物理を取った人だけが

  • taking so long on each problem.

    頭の中にコンピューターを持てるかのようです 問題を解くのに時間のかかるあなたは

  • It's not that different with linear algebra, and luckily, just as with trigonometry, there are a


  • handful of intuitionsvisual intuitionsunderlying much of the subject. And unlike the trig example,

    これは線型代数でも同じです 幸運なことに,三角関数と同じく,線型代数も

  • the connection between the computation and these visual intuitions is typically pretty

    多くのことを直感的に「視覚的に」理解できます 三角関数の例とちがい,線型代数は

  • straightforward. And when you digest these, and really understand the relationship between the

    数値計算と視覚的直感のつながりを とても理解しやすいです

  • geometry and the numbers, the details of the subject, as well as how it's used in practice, start to

    そしてこれらを身につけ 幾何学的意味と数値計算の関係を

  • feel a lot more reasonable.

    本当に理解できたとき (実際どのように使えばよいかも含めて)

  • In fairness, most professors do make an effort to convey that geometric understanding; the sine


  • example is a little extreme, but I do think that a lot of courses have students spending a

    多くの教授は,幾何学的に理解してもらおうと 努力しています

  • disproportionate amount of time on the numerical side of things, especially given that in this day

    サイン関数の例は少し極端ですが 多くの授業では,数値計算に

  • and age, we almost always get computers to handle that half, while in practice, humans worry about

    必要以上に時間を費やしています 特に,パソコンのある現代では

  • the conceptual half.

    パソコンによる計算に半分を費やし 人間がその概念的な理解に

  • So this brings me to the upcoming videos. The goal is to create a short, binge-watchable series


  • animating those intuitions, from the basics of vectors, up through the core topics that make up the

    だからわたしはこれらのビデオを作りました ゴールは,短くて手軽にみれるシリーズをつくることです

  • essence of linear algebra. I'll put out one video per day for the next five days, then after that,

    視覚的かつ直感的に ベクトルの基本から線型代数のコアとなる話題を

  • put out a new chapter every one to two weeks. I think it should go without saying that you cannot

    説明します 1日ひとつのビデオを5日間でアップし

  • learn a full subject with a short series of videos, and that's just not the goal here, but what you

    その後,1週間から2週間おきにビデオをアップします 短いビデオシリーズで

  • can do, especially with this subject, is lay down all the right intuitions, so that the learning you

    すべてを学ぶことはできませんが それが目標でもありません

  • do moving forward is as productive and fruitful as it can be. I also hope this can be a resource for


  • educators whom are teaching courses that assume fluency with linear algebra, giving them a place to

    生産的で実りのある学習をしてもらうことです また,これらは先生たちの教材にもなるでしょう

  • direct students whom need a quick brush-up.

    線型代数を既習とする授業で ざっとした復習が

  • I'll do what I can to keep things well-paced throughout, but it's hard to simultaneously account for


  • different people's different backgrounds and levels of comfort, so I do encourage you to readily

    できるだけよいペースで進めていきますが さまざまな人の

  • pause and ponder if you feel that it's necessary. Actually, I'd give that same advice when watching

    さまざまな背景や理解のレベルに 同時に合わせるのは困難です そのため必要な時は

  • any math video, even if it doesn't feel too quick, since the thinking that you do in your own time

    ビデオを止めて考えることを 強くおすすめします 実際,どんな数学のビデオでも

  • is where all the learning really happens, don't you think?

    止めて考えることをおすすめします たとえ速すぎると感じないときもです なぜなら,自分のペースで考えることは

  • So, with that as an introduction, I'll see you in the next video.

    本当に学習するということだからです そう思いませんか?

  • Captioned by Navjivan Pal Reviewed by Johann Hemmer 07/08/16

    では,イントロダクションはこの辺で 次のビデオで会いましょう!

00:00:08,095 --> 00:00:11,635 Hey, everyone! So I'm pretty excited about the next sequence of videos that I'm doing. It'll be about

線型代数ほど基本的な理論はない 教授や教科書が途方もない行列計算で その簡潔さを難しく見せているにも関わらず ――ジャン・デュドネ


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B1 中級 日本語 計算 幾何 関数 理解 数値 ビデオ

線形代数のエッセンス プレビュー (Essence of linear algebra preview)

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    jeffrey に公開 2021 年 01 月 14 日