It'll be about linear algebra, which—as a lot of you know—is 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 noticed—generally 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 products—which 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，

determinant—or 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, but—roughly speaking—the 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 intuitions—visual intuitions—underlying 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，