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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

00:00:08,095 --> 00:00:11,635 嘿大家！我很期待我接下來要推出的一系列影片，

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

它是有關線性代數。你們很多人可能已經知道，

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

線性代數幾乎是所有科學學門都需要的科目。

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

然而我同時也注意到，第一次學習它的學生常不容易理解它。

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

學生可能上完一整堂相關課程後學到如何算很多事情，

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

比如矩陣乘法，或行列式，或叉積(利用行列式計算)，或特徵值。

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

但他們可能沒有真正理解為什麼矩陣乘法要這麼定義，

• determinant, or what an eigenvalue really represents.

為什麼向量叉積和行列式有關，

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

或著特徵值真正代表甚麼。

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

通常，學生到最後很會做矩陣的數值運算，

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

但只模糊地知道它背後的幾何直覺。

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

但是，有很重要的差別存在於以數值運算來理解線性代數，

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

和以幾何理解線性代數。兩種都有其重要性。

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

不過，粗淺來說，幾何理解能讓你判斷要用何種工具來解決某個的問題，

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

感受到為什麼這些工具有用，和知道如何解釋這些結果，

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

而對其數值運算的了解，則讓你能實際地應用這些工具。

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

那麼，假如你學習線性代數的時候沒有堅實地學得其幾何意義，

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

問題可能會一直隱藏在那，直到你已經深入你探究的領域時才顯露出來，

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

不管那是計算機科學，工程學，統計學，經濟學，甚至是數學本身。

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

一旦你在一堂課上，或是一個工作上，其要求對線性代數很熟悉，

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

你的教授，或你的同事使用線性代數的方法可能看起來會很神奇。

• crunching all the numbers in their head.

他們會很快知道要選用哪個工具，還有答案大概會是甚麼樣子，

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

在某種程度上，看起像是種計算的巫術，

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

如果你假定他們是真的在腦子裡做數字運算的話。

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

作為一個比喻，想像你第一次學習三角函數裡的正弦函數，

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

你看到這個無限的多項式。順便說一句，這是你的計算器如何計算正弦函數的方式。

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

作業裡，您可能會被要求練習計算正弦函數的近似值，

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

藉由在公式中插入各個數目，然後在某個合理的點停止。

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

然後，公平的說，假設你有一個模糊的概念，覺得三角函數應該和三角形有關，

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

但從不真正知道它們如何相關，而這也不是那堂課程的重點。

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

之後，如果你修了一門物理課，課中到處要用正弦函數和餘弦函數，

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

而其他人能很快地使用它們，而且約略知道某值的正弦函數值是多少，

• taking so long on each problem.

這將會非常嚇人的，不是嗎？

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

這會看起來像是能了解物理的人只有那些大腦像電腦的人，

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

而你會覺得自己又慢又笨，才需要對每一個問題花這麼久時間。

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

線性代數和這沒甚麼太大的不同。而且，幸運的是，正如同三角函數，

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

其中只有為數可數的直覺- 可視覺化直覺-

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

存在於這個學科大多數的地方。而不同於三角函數，

• feel a lot more reasonable.

線性代數的計算與這些視覺化直覺之間的連結是相當直接的。

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

當你消化這些，而且真正理解其幾何和計算層面間的關係，

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

這個學科的細節，以及它實際上如何使用，就會開始顯得更合理。

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

憑心而論，多數教授都有努力傳達線性代數的幾何意義。

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

正弦函數的例子是有點極端。

• the conceptual half.

但我認為，很多課程讓學生花不成比例的時間在計算上，

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

尤其是考慮到在這個時代，我們幾乎都讓電腦處理計算的那一半，

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

而實際上，人們更重視概念性的一半。

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

因此，這使我製作出接下來想的影片。我的目標是製作出一個簡短的，很有看頭的系列，

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

以動畫呈現那些直覺，從最基本的向量，到一些核心主題，

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

其組成了線性代數的精華。接下來的五天，我會每天推出一個視頻，

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

在這之後，每隔一到兩週推出一個新的章節。

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

不用說，你沒辦法用一系列簡短的影片學完科目的全部，而這也不是我的目的。

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

不過你能做到的，尤其是在這個科目上，是形成所有正確的直覺，

• direct students whom need a quick brush-up.

來讓你的學習過程盡可能地有效果地進展。

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

我也希望這能成為一個資源給老師們，讓他們在教需要很了解線性代數的課程時，

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

有個地方給學生快速的複習一下。

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

我會盡可能的讓內容進展的速度剛剛好，

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

但我很難同時考慮不同人的不同背景和理解程度，

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

所以我鼓勵你不時暫停，然後想一想，如果你覺得有必要。

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

其實，我給看任何數學影片的人同樣的建議，

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

即使它感覺起來不會太快，因為你用你自己的時間思考的時候，

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

00:00:08,095 --> 00:00:11,635 嘿大家！我很期待我接下來要推出的一系列影片，

B1 中級 中文 美國腔 代數 線性 函數 幾何 計算 直覺

線性代數精髓預覽 (Essence of linear algebra preview)

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jeffrey 發佈於 2021 年 01 月 14 日