跟读练习: Essence of linear algebra preview - 通过视频学习英语口语
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Hey everyone!
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So, I'm pretty excited about the next sequence of videos that I'm doing.
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They'll be about linear algebra, which as a lot of you know, is one of those subjects that's required knowledge for just about any technical discipline, but it's also, I've noticed,
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generally poorly understood by students taking it for the first time time.
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A student might go through a class and learn how to compute lots of things, like matrix multiplication, or the determinant, or cross products which use the determinant,
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or eigenvalues, but they might come out without really understanding why matrix multiplication is defined the way that it is,
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why the cross product has anything to do with the determinant, or what an eigenvalue really represents.
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Oftentimes, students end up well-practiced in the numerical operations of matrices, but are only vaguely aware of the geometric intuitions underlying it all.
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But there's a fundamental difference between understanding linear algebra on a numeric level and understanding it on a geometric level.
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Each has its place, but roughly speaking, the geometric understanding is what lets you judge what tools to use to solve specific problems,
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feel why they work, and know how to interpret the results.
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And the numeric understanding is what lets you actually carry through the application of those tools.
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Now, if you learn linear algebra without getting a solid foundation in that geometric understanding, the problems can go unnoticed for a while,
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until you've gone deeper into whatever field you happen to pursue, whether that's computer science, engineering, statistics, economics, or even math itself.
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Once you're in a class, or a job for that matter, that assumes fluency with linear algebra,
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the way that your professors or your co-workers apply that field could seem like utter magic.
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They'll very quickly know what the right tool to use is, and what the answer roughly looks like in a way
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that would seem like computational wizardry if you assume that they're actually crunching all the numbers in their head.
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Here, as an analogy, imagine that when you first learned about the sine function in trigonometry, you were shown this infinite polynomial.
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This, by the way, is how your calculator evaluates the sine function.
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For homework, you might be asked to practice computing approximations of the sine function, by plugging in various numbers to the formula and cutting it off at a reasonable point.
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And in fairness, let's say you had a vague idea that this was supposed to be related to triangles, But exactly how had never really been clear, and was just not the focus of the course.
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Later on, if you took a physics course, where sines and cosines are thrown around left and right, and people are able to tell pretty immediately how to apply them
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and roughly what the sign of a certain value will be, it would be pretty intimidating, wouldn't it?
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It would make it seem like the only people who are cut out for physics are those with computers for brains, and you would feel unduly slow or dumb for taking so long on each problem.
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It's not that different with linear algebra.
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And luckily, just as with trigonometry, there are a handful of intuitions, visual intuitions, underlying much of the subject.
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And unlike the trig example, the connection between the computation and these visual intuitions is typically pretty straightforward.
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And when you digest these, and really understand the relationship between the geometry and the numbers, the details of the subject, as well as how it's used in practice,
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start to feel a lot more reasonable.
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In fairness, most professors do make an effort to convey that geometric understanding.
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The science example is a little extreme.
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But I do think
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that a lot of courses have students spending a disproportionate amount of time on the numerical side of things, especially given that in this day and age,
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we almost always get computers to handle that half, while in practice, humans worry about the conceptual half.
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So this brings me to the upcoming videos.
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The goal is to create a short, binge-watchable series animating those intuitions from the basics of vectors up through the core topics
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that make up the essence of linear algebra.
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I'll put out one video per day for the next five days, then after that put out a new chapter every one to two weeks.
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I think it should go without saying that you cannot learn a full subject with a short series of videos, and that's just not the goal here.
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But what you can do, especially with this subject, is lay down all the right intuitions
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so the learning that you do moving forward is as productive and fruitful as it can be.
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I also hope this can be a resource for educators who are teaching courses that assume fluency with linear algebra, giving them a place to direct students that need a quick brush up.
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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,
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so I do encourage you to readily pause and ponder if you feel that it's necessary.
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Actually, I'd give that same advice for watching any math video, even if it doesn't feel too quick, since the thinking that you do on your own time is where all the learning really happens, don't you think?
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So, with that as an introduction, I'll see you next video.
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本课的词汇与口语要点
这节 C1 级别的口语课以视频“Essence of linear algebra preview”为素材。 视频中反复出现的词有：algebra, linear, understanding, intuitions, geometric。 这段视频共有 47 个句子、917 个单词可供跟读。 讲话部分时长为 4:43。 说话人语速较快，每分钟约 195 个词，会出现较多连读和弱读。 只有 81% 的单词属于英语最常用的 3,000 词，词汇难度较高。
视频中的重点词汇
视频中最难的 15 个单词，附发音和释义：
| 单词 | 发音 | 释义 |
|---|---|---|
| algebra 名词 | /ˈæl.d͡ʒɪ.bɹə/ | 代數 /代数 |
| intuition 名词 | /ˌɪn.tjuːˈɪʃ.ən/ | 直覺 /直觉, 直感 |
| sine 名词 | /saɪn/ | 正弦 |
| determinant 名词 | /dɪˈtɜːmɪnənt/ | 決定(性)因素 /决定(性)因素 |
| compute 动词 | /kəmˈpjuːt/ | 計算 /计算 |
| eigenvalue 名词 | /ˈaɪ.ɡənˌvæl.ju/ | 特徵值 /特征值, 本徵值 /本征值 |
| fluency 名词 | /ˈfluːənsi/ | 流暢 /流畅 |
| multiplication 名词 | /ˌmʌltɪplɪˈkeɪʃən/ | 乘法 |
| trigonometry 名词 | /ˌtɹɪɡəˈnɒmətɹi/ | 三角法, 三角學 /三角学 |
| numerical 形容词 | /n(j)uːˈmɛɹ.ɪ.kəl/ | 數值的 /数值的 |
| animate 形容词 | /ˈæn.ə.mət/ | 活的, 有生命的 |
| cosine 名词 | /ˈkoʊ.saɪn/ | 餘弦 /余弦 |
| intimidate 动词 | /ɪnˈtɪmɪdeɪt/ | 威脅 /威胁, 恐嚇 /恐吓 |
| polynomial 名词 | /ˌpɑliˈnoʊmiəl/ | 多項式 /多项式 |
| ponder 动词 | /ˈpɑn.dəɹ/ | 思考, 琢磨 |
视频中出现的短语动词
| 单词 | 发音 | 释义 |
|---|---|---|
| come out 动词 | /ˌkʌm ˈaʊt/ | 出, 出來 /出来 |
| end up 动词 | 到达 | |
| go through 动词 | 通過 /通过 | |
| lie down 动词 | 偃, 躺下 | |
| make up 动词 | /ˌmeɪk ˈʌp/ | 組成 /组成 |
视频中的语法
说话人最常用的结构，并附上视频中的原话：
| 结构 | 视频中的用法 |
|---|---|
| 被动语态 be + 过去分词 — 强调发生了什么，而不是谁做的 | is defined · were shown · might be asked |
| 现在完成时 have/has + 过去分词 — 过去发生但与现在仍有关联的事 | I've noticed · you've gone |
需要注意的发音
说话人用了 15 次缩略和弱读形式，例如 I'll, I'm, they'll。请按听到的简短形式来说。
- “sh” 和 “zh” 音: intuition /ˌɪn.tjuːˈɪʃ.ən/, multiplication /ˌmʌltɪplɪˈkeɪʃən/, approximation /əˌpɹɑk.sɪˈmeɪ.ʃən/, computation /ˌkɑmpjʊˈteɪʃn̩/, disproportionate /dɪs.pɹəˈpɔː.ʃə.nət/
- 长单词——注意重音位置: intuition /ˌɪn.tjuːˈɪʃ.ən/, geometric /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/, determinant /dɪˈtɜːmɪnənt/, eigenvalue /ˈaɪ.ɡənˌvæl.ju/, multiplication /ˌmʌltɪplɪˈkeɪʃən/
中文母语者容易读错的音：
- /v/ — 不要读成 /w/，上齿轻触下唇: eigenvalue /ˈaɪ.ɡənˌvæl.ju/, vaguely /ˈveɪɡli/, convey /kənˈveɪ/, vague /veɪɡ/, vector /ˈvɛktɚ/
- 词尾辅音 — 要读清楚，后面不要加元音: geometric /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/, determinant /dɪˈtɜːmɪnənt/, matrix /ˈmeɪ.tɹɪks/, compute /kəmˈpjuːt/, numeric /n(j)uːˈmɛɹɪk/
- /r/ 和 /l/ 的区别: algebra /ˈæl.d͡ʒɪ.bɹə/, numerical /n(j)uːˈmɛɹ.ɪ.kəl/, fruitful /ˈfɹuːtfl̩/, triangle /ˈtɹaɪˌæŋɡəl/, readily /ˈɹɛdɪli/
如何用这段视频练习
- 先完整听一遍视频，不要开口，记下不认识的单词。
- 先用 0.75 倍速逐句跟读，熟练之后再回到正常速度。
- 录下自己的声音并与原声对比，特别注意 algebra, intuition, sine 这类单词。
什么是跟读法？
跟读法 (Shadowing) 是一种有科学依据的语言学习技巧，最初开发用于专业口译员的培训，并由多语言者Alexander Arguelles博士普及。这个方法简单而强大：您在听英语母语原声的同时立即大声重复——就像是一个延迟1-2秒紧跟说话者的影子。与被动听力或语法练习不同，跟读法强迫您的大脑和口腔肌肉同时处理并模仿真实的讲话模式。研究表明它能显着提高发音准确性，语调，节奏，连读，听力理解和口语流利度——使其成为雅思口语备考和真实英语交流最有效的方法之一。