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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,

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B1 中級 中文 美國腔 代數 線性 函數 幾何 計算 直覺

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    jeffrey 發佈於 2021 年 01 月 14 日
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