シャドーイング練習: Vectors | Chapter 1, Essence of linear algebra - 動画で英語スピーキングを学ぶ
レッスンを作成中...
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Music The fundamental, root of it all,
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building block for linear algebra is the vector.
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So it's worth making sure that we're all on the same page about what exactly a vector is.
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You see, broadly speaking, there are three distinct but related ideas about vectors, which I'll call the physics student perspective,
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the computer science student perspective, and the mathematician's perspective.
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The physics student perspective is that vectors are arrows pointing in space.
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What defines a given vector is its length and the direction it's pointing, but as long as those two facts are the same, you can move it all around and it's still the same vector.
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Vectors that live in the flat plane are two-dimensional, and those sitting in broader space that you and I live in are three-dimensional.
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The computer science perspective is that vectors are ordered lists of numbers.
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For example, let's say you were doing some analytics about house prices, and the only features you cared about were square footage and price.
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You might model each house with a pair of numbers, the first indicating square footage and the second indicating price.
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Notice the order matters here.
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In the lingo, you'd be modeling houses as two-dimensional vectors, where in this context, vector is pretty much just a fancy word for list,
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and what makes it two-dimensional is the fact that the length of that list is two.
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The mathematician, on the other hand, seeks to generalize both these views, basically saying that a vector can be anything where there's a sensible notion of adding two vectors
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and multiplying a vector by a number, operations that I'll talk about later on in this video.
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The details of this view are rather abstract, and I actually think it's healthy to ignore it until the last video of this series, favoring a more concrete setting in the interim.
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But the reason I bring it up here is
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that it hints at the fact that the ideas of vector addition and multiplication by numbers
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will play an important role throughout linear algebra.
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But before I talk about those operations, let's just settle in on a specific thought to have in mind when I say the word vector.
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Given the geometric focus that I'm shooting for here, whenever I introduce a new topic involving vectors, I want you to first think about an arrow,
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and specifically, think about that arrow inside a coordinate system, like the xy-plane, with its tail sitting at the origin.
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This is a little bit different from the physics student perspective, where vectors can freely sit anywhere they want in space.
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In linear algebra, it's almost always the case that your vector will be rooted at the origin.
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Then, once you understand a new concept in the context of arrows in space, we'll translate it over to the list of numbers point of view, which we can do by considering the coordinates of the vector.
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Now, while I'm sure that many of you are already familiar with this coordinate system, it's worth walking through explicitly,
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since this is where all of the important back and forth happens between the two perspectives of linear algebra.
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our attention on two dimensions for the moment, you have a horizontal line, called the x-axis, and a vertical line, called the y-axis.
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The place where they intersect is called the origin, which you should think of as the center of space and the root of all vectors.
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After choosing an arbitrary length to represent one, you make tick marks on each axis to represent this distance.
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When I want to convey the idea of 2D space as a whole, which you'll see comes up a lot in these videos, I'll extend these tick marks to make grid lines,
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but right now they'll actually get a little bit in the way.
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The coordinates of a vector is a pair of numbers
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that basically gives instructions for how to get from the tail of that vector at the origin to its tip.
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The first number tells you how far to walk along the x-axis, positive numbers indicating rightward motion, negative numbers indicating leftward motion,
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and the second number tells you how far to walk parallel to the y-axis after that, positive numbers indicating upward motion and negative numbers indicating downward motion.
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To To distinguish vectors from points, the convention is to write this pair of numbers vertically with square brackets around them.
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Every pair of numbers gives you one and only one vector, and every vector is associated with one and only one pair of numbers.
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What about in three dimensions?
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Well, you add a third axis, called the z-axis, which is perpendicular to both the x and y axes,
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and in this case, each vector is associated with an ordered triplet of numbers.
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The first tells you how far to move along the x-axis, the second tells you how far to move parallel to the y-axis,
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and the third one tells you how far to then move parallel to this new z-axis.
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Every triplet of numbers gives you one unique vector in space, and every vector in space gives you exactly one triplet of numbers.
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Alright, so back to vector addition and multiplication by numbers.
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After all, every topic in linear algebra is going to center around these two operations.
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Luckily, each one's pretty straightforward to define.
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Let's say we have two vectors, one pointing up and a little to the right, and the other one pointing right and down a bit.
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To add these two vectors, move the second one so that its tail sits at the tip of the first one.
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Then, if you draw a new vector from the tail of the
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first one to where the tip of the second one now sits, that new vector is their sum.
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This definition of addition, by the way, is pretty much the only time in linear algebra where we let vectors stray away from the origin.
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Now why is this a reasonable thing to do?
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Why this definition of addition and not some other one?
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Well the way I like to think about it is that each vector represents a certain movement, a step with a certain distance and direction in space.
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If you take a step along the first vector, then take a step in the direction and distance described by the second vector,
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the overall effect is just the same as if you moved along the sum of those two vectors to start with.
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You could think about this as an extension of how we think about adding numbers on a number line.
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One way that we teach kids to think about this, say with 2 plus 5, is to think of moving two steps to the right, followed by another five steps to the right.
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The overall effect is the same as if you just took seven steps to the right.
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In fact, let's see how vector addition looks numerically.
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The first vector here has coordinates , and the second one has coordinates .
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When you take the vector sum using this tip-to-tail method, you can think of a four-step path from the origin to the tip of the second vector.
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Walk one to the right, then two up, then three to the right, then one down.
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Reorganizing these steps so that you first do all of the rightward motion, then do all the vertical motion, you can read it as saying,
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first move to the right, then move up up.
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So the new vector has coordinates 1 plus 3 and 2 plus negative 1.
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In general, vector addition in this list of numbers conception looks like matching up their terms and adding each one together.
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The other fundamental vector operation is multiplication by a number.
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Now this is best understood just by looking at a few examples.
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If you take the number 2 and multiply it by a given vector, it means you stretch out that vector so that it's two times as long as when you started.
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If you multiply that vector by, say, one-third, it means you squish it down so that it's one-third the original length.
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When you multiply it by a negative number, like negative 1.8, then the vector first gets flipped around, then stretched out by that factor of 1.8.
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This process of stretching or squishing or sometimes reversing the direction of a vector is called scaling.
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And whenever you catch a number, like 2 or 1 third or negative 1.8, acting like this, scaling some vector, you call it a scalar.
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In fact, throughout linear algebra, one of the main things that numbers do is scale vectors.
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So it's common to use the word scalar pretty much interchangeably with the word number.
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Numerically, stretching out a vector by a factor of, say, 2, corresponds with multiplying each of its components by that factor, 2.
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So in the conception of vectors as lists of numbers, multiplying a given vector by a scalar means multiplying each one of those components by that scalar.
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You'll see in the following videos what I mean when I say
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that linear algebra topics tend to revolve around these two fundamental operations, vector addition and scalar multiplication.
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And I'll talk more in the last video about how and why the mathematician thinks only about these operations, independent and abstracted away from however you choose to represent vectors.
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In truth, it doesn't matter whether you think about vectors as fundamentally being arrows in space, like I'm suggesting you do, that happen to have a nice numerical representation,
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or fundamentally as lists of numbers that happen to have a nice geometric interpretation.
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The usefulness of linear algebra has less to do with either
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one of these views than it does with the ability to translate back and forth between them.
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It gives the data analyst a nice way to conceptualize many lists of numbers in a visual way,
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which can seriously clarify patterns in data and give a global view of what certain operations do.
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And on the flip side, it gives people like physicists and computer graphics programmers a language to describe space and the manipulation of space,
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using numbers that can be crunched and run through a computer computer.
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When I do mathy animations, for example, I start by thinking about what's actually going on in space and then get the computer to represent things numerically,
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thereby figuring out where to place the pixels on the screen.
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And doing that usually relies on a lot of linear algebra understanding.
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So there are your vector basics
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and in the next video I'll start getting into some pretty neat concepts surrounding vectors like span, bases, and linear dependence.
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See you then!
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このレッスンの語彙とスピーキングのポイント
繰り返し出てくる語は次のとおりです：vector, space, linear, algebra, axis。 この動画には、シャドーイング用の文が97文、単語が1795語あります。 音声の長さは9:36です。 話す速さは速く、1分あたり約187語です。音のつながりや弱く発音される音が多くなります。 英語の頻出3,000語に含まれる単語は79%だけなので、語彙は難しめです。
この動画の重要語彙
動画の中で特に難しい単語15語を、発音と意味つきで紹介します。
| 単語 | 発音 | 意味 |
|---|---|---|
| vector 名詞 | /ˈvɛktɚ/ | ベクトル |
| algebra 名詞 | /ˈæl.d͡ʒɪ.bɹə/ | 代数 |
| axis 名詞 | /ˈæksɪs/ | 軸 |
| multiply 動詞 | /ˈmʌltɪplaɪ/ | 掛ける |
| scalar 形容詞 | /ˈskeɪ.lə/ | スカラー |
| multiplication 名詞 | /ˌmʌltɪplɪˈkeɪʃən/ | 掛け算, 乗法 |
| dimensional 形容詞 | /daɪˈmɛn.ʃə.nəl/ | -次元の |
| triplet 名詞 | /ˈtɹɪplət/ | 三つ組 |
| mathematician 名詞 | /ˌmæθ.(ə.)məˈtɪʃ.ən/ | 数学者 |
| conception 名詞 | /kənˈsɛpʃən/ | 妊娠, 受胎 |
| tick 名詞 | /tɪk/ | だに, ごさらぎ |
| translate 動詞 | /tɹænzˈleɪt/ | 訳す, 翻訳する |
| dimension 名詞 | /daɪˈmɛn.ʃən/ | 面積, 寸法 |
| perpendicular 形容詞 | /ˌpɜː.pənˈdɪk.jə.lə(ɹ)/ | 垂直の, 縦の |
| physicist 名詞 | /ˈfɪzɪsɪst/ | 物理学者 |
動画に出てくる句動詞
| 単語 | 意味 |
|---|---|
| come up 動詞 | 上がる, 登る |
注意したい発音
話し手はI'll, I'm, you'llなど、短縮形や弱形を15回使っています。聞こえたとおりの短い形で発音しましょう。
- 「th」の音: mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/, thereby /ðɛɹˈbaɪ/
- 「sh」と「zh」の音: multiplication /ˌmʌltɪplɪˈkeɪʃən/, dimensional /daɪˈmɛn.ʃə.nəl/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/, conception /kənˈsɛpʃən/, dimension /daɪˈmɛn.ʃən/
- 長い単語（アクセントの位置に注意）: coordinate /koʊˈɔɹ.dəˌneɪt/, multiplication /ˌmʌltɪplɪˈkeɪʃən/, dimensional /daɪˈmɛn.ʃə.nəl/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/, geometric /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/
日本語話者が苦手な音：
- /r/ と /l/ の区別 — /l/ は舌先を歯茎につけ、/r/ はどこにもつけない: algebra /ˈæl.d͡ʒɪ.bɹə/, triplet /ˈtɹɪplət/, translate /tɹænzˈleɪt/, generalize /ˈd͡ʒɛn.(ə.)ɹə.laɪz/, perpendicular /ˌpɜː.pənˈdɪk.jə.lə(ɹ)/
- /v/ — /b/ にならないように、上の歯を下唇に当てる: vector /ˈvɛktɚ/, revolve /ɹɪˈvɑlv/, vertically /ˈvɝ.tɪ.kəli/, convey /kənˈveɪ/
- /f/ — 「フ」ではなく、上の歯と下唇で出す: physicist /ˈfɪzɪsɪst/, usefulness /ˈjuːsfəlnəs/, straightforward /ˌstɹeɪtˈfɔː(ɹ)wə(ɹ)d/, clarify /ˈklæɹ.ɪ.faɪ/, freely /ˈfɹili/
この動画での練習方法
- まず声を出さずに動画を最後まで聞き、知らない単語をメモします。
- まず0.75倍速で一文ずつシャドーイングし、慣れてきたら通常の速度に戻します。
- 自分の声を録音して元の音声と比べます。vector, algebra, axisなどの単語に特に注意しましょう。
シャドーイングとは？英語上達に効果的な理由
シャドーイング（Shadowing）は、もともとプロの通訳者養成プログラムで開発された言語学習法で、多言語習得者として知られるDr. Alexander Arguelles によって広く普及されました。方法はシンプルですが非常に効果的：ネイティブスピーカーの英語を聞きながら、1〜2秒の遅延で声に出してすぐに繰り返す——まるで「影（shadow）」のように話者を追いかけます。文法ドリルや受動的なリスニングと異なり、シャドーイングは脳と口の筋肉が同時にリアルタイムで英語を処理・再現することを強制します。研究により、発音精度、抑揚、リズム、連音、リスニング力、そして会話の流暢さが大幅に向上することが確認されています。IELTSスピーキング対策や自然な英語コミュニケーションを目指す方に特におすすめです。