—Hermann Weyl

--HermannWeyl

The fundamental, root-of-it-all building block for linear algebra is the vector, so it's

綫性代數最基本的，構建一切最根本的磚塊就是向量，

worth

所以我們對「向量到底是什麽」都有一樣的理解將十分重要。

making sure that we're all on the same page about what exactly a vector is.

你們知道，大緻的來講對向量有三種獨特但相關的看法，

You see, broadly

我稱他們為物理學生的角度

speaking there are three distinct but related ideas about vectors, which I'll call the physics

電腦科學學生的和數學家的角度。

student perspective, the computer science student perspective, and the mathematician's

物理學生認為向量是指向空間的一些箭。

perspective.

定義一個向量需要它的長度，和它所指著的方向，

The physics student perspective is that vectors are arrows pointing in space.

而只要這兩個值都不變，你可以把它四處移動而仍是同一個向量。

What defines a given

在一個平面中的那些向量是2-維的，

vector is its length, and the direction it's pointing in, but as long as those two facts

而那些在你我所生活的空間裏的是3-維的。

are the

從計算科學的角度來看向量衹是有次序列在表裏的一些數字。

same, you can move it all around and it's still the same vector.

舉個例子，假定說你對房子的價格作些分析，

Vectors that live in the flat plane

而你所感興趣的只是平方尺數和價格。

are two-dimensional, and those sitting in broader space that you and I live in are three-dimensional.

你也許把每座房當作一對數字：

The computer science perspective is that vectors are ordered lists of numbers.

第一個指出平方尺數，而第二個指出價格。

For example, let's

注意在這裏的次序是重要的。

say that you were doing some analytics about house prices, and the only features you cared

用術語來講，你把房子模擬成2-維的向量，

about

而在這個上下文意義上「向量」就不過是「列表」的一種花漂的說法，

were square footage and price.

而2-維的向量，就是指那張表的長度是2

You might model each house with a pair of numbers: the first

另一方面，數學家追求對這兩種看法的通用化，

indicating square footage, and the second indicating price.

基本上就是說一個向量可以是任何東西，

Notice that the order matters here.

只要保證兩個向量加起來和矢量與常數相乘是有意義的即可

In the lingo, you'd be modelling houses as two-dimensional vectors, where in this context,

我將在這個視頻說說這兩種運算。

"vector" is pretty much just a fancy word for "list", and what makes it two-dimensional

這種看法的具體細節是相當抽象的，

is the fact

我想把這個留待這個系列最後兩集。

that the length of that list is 2.

而在這期間打好一個更堅實的基礎。

The mathematician, on the other hand, seeks to generalise both of these views, basically

我在這提起這種觀點的理由，是它暗示了事實上，

saying that

向量的相加和向量乘以數字，

a vector can be anything where there's a sensible notion of adding two vectors, and multiplying

將在整個線性代數裏起着一個重要的作用。

a

但在我講這些運算之前，

vector by a number, operations that I'll talk about later on in this video.

我們先來確立思考「向量」的特別方式。

The details of this view

我打算在這裏集中在幾何上，

are rather abstract, and I actually think it's healthy to ignore it until the last video

每當我引進一個涉及向量的新題目的時候，

of this

我要你第一步先想到一個箭頭，

series, favoring a more concrete setting in the interim,

準確來說，想像一個直角座標內的箭頭，

but the reason that I bring it up here is that it hints at the fact that ideas of vector

那支箭的箭尾就在原點上。

addition

這和物理學生的角度是有點不同，

and multiplication by numbers will play an important role throughout linear algebra.

因為他們眼中的向量可以自由地放在空間裏任何地方的。

But before I talk about those operations, let's just settle in on a specific thought

在線性代數裏，向量的起點幾乎總是在原點。

to have in mind

然後，一旦你理解了這種在空間裏的箭頭的新概念，

when I say the word "vector".

我們將把它轉成表中一些數字，

Given the geometric focus that I'm shooting for here, whenever I

就可以通過考慮這向量的坐標來達成。

introduce a new topic involving vectors, I want you to first think about an arrow—and

現在我相信你們中許多人都熟悉這個坐標系統的同時，

specifically,

這還是值得一步一步來看一下，

think about that arrow inside a coordinate system, like the x-y plane, with its tail

因爲這正是在兩種角度中來來回回的線性代數。

sitting at the origin.

我們先集中於2-維空間內，

This is a little bit different from the physics student perspective, where vectors can freely

你有一條水平綫，叫做x軸，和一條垂直的綫，叫做y軸。

sit

它們相交的地方叫做原點，

anywhere they want in space.

你應該把它想像成空間的中心和所有矢量的起點。

In linear algebra, it's almost always the case that your vector will be

選一個任意的長度來代表1之後，你就可以在兩個軸上標上刻度來代表這個距離。

rooted at the origin.

如果我想在影片中表達整個2-維空間時，

Then, once you understand a new concept in the context of arrows in space,

我將延伸這些刻度做出網格綫，

we'll translate it over to the list-of-numbers point-of-view, which we can do by considering

但它們現在還不需要出現。

the coordinates of the vector.

一個向量的坐標是一組數字，

Now while I'm sure that many of you are familiar with this coordinate system, it's worth walking

基本上指出怎樣把箭尾﹙原點上﹚移到箭頭上去的。

through explicitly, since this is where all of the important back-and-forth happens between

第一個數字告訴你在x軸上該移動多遠：正數指向右移動，負數指向左移動；

the two

而第二個數字告訴你第一次移動後平行y軸要移動多遠：

perspectives of linear algebra.

正數指向上移動，而負數指向下移動。

Focusing our attention on two dimensions for the moment, you have a

區別向量與點的慣用方法，

horizontal line, called the x-axis, and a vertical line, called the y-axis.

是把這一組數竪寫並用方括號把它們括起來。

The place where they

每一組數字表示一個，且只有一個向量，

intersect is called the origin, which you should think of as the center of space and

而每個向量只被一個，且只有一組數字代表。

the root of all vectors.

那麼 3-維向量是什麽呢？

After choosing an arbitrary length to represent 1, you make tick-marks on each axis to

在直角座標加上第三個軸，叫做z軸，

represent this distance.

它同時和x和y軸垂直，

When I want to convey the idea of 2-D space as a whole, which you'll see

則這種情況下，每個向量會和一個有次序的三個數字對應：

comes up a lot in these videos, I'll extend these tick-marks to make grid-lines, but right

第一個告訴你沿著x軸移動多遠

now

第二個數字告訴你和y軸平行地移動多遠，

they'll actually get a little bit in the way.

而第三個數字則告訴你要和z-軸平行地移動多遠。

The coordinates of a vector is a pair of numbers that

每一組三個數字給你一個空間中獨特的矢向量，

basically give instructions for how to get from the tail of that vector—at the origin—to

而每一個在空間中的向量剛好給你一組三個數字。

its tip.

所以回到向量的加法，以及和一些數字的乘法。

The first number tells you how far to walk along the x-axis—positive numbers indicating

畢竟，在線性代數中每一個主題都將環繞著這兩種運算的。

rightward

好在，每一種都很直接了當地定義。

motion, negative numbers indicating leftward motion—and the second number tell you how

比方說我們有兩個向量，一個朝上指，偏一點右，

far to walk

而另一個朝右指，而偏下一點。

parallel to the y-axis after that—positive numbers indicating upward motion, and negative

要把這兩個向量相加，移動第二個使它的箭尾放到第一個向量的箭頭；

numbers

然後如果你從第一個的箭尾到第二個的箭頭現在的地方畫一個向量，

indicating downward motion.

這個新的向量就是它們的和。

To distinguish vectors from points, the convention is to write this pair

順帶一提，加法的定義是在線性代數裏基本是唯一一次我們讓向量從原點偏離。

of numbers vertically with square brackets around them.

而現在為什麽要這樣定義加法呢？而不是其他定義呢？

Every pair of numbers gives you one and only one vector, and every vector is associated

這裏我喜歡想像每一個向量代表著一種移動，

with one and

也就是在空間裏以給定的距離和方向的踏出一步。

only one pair of numbers.

如果你沿著第一個向量走一步，

What about in three dimensions?

然後按第二個向量所描述的方向和距離移動，

Well, you add a third axis, called the z-axis,

總效果就和你一開始就沿著這兩個矢量的和走是一樣的。

which is perpendicular to both the x- and y-axes, and in this case each vector is associated

你也可以把它當作怎樣在一根數軸上把一些數字加起來的延伸。

with an ordered triplet of numbers: the first tells you how far to move along the x-axis,

我們教孩子來想的一個方法，

the second

比方說2 + 5，就是向右走二步，接著向右另外5 步。

tells you how far to move parallel to the y-axis, and the third one tells you how far

這總效果和你一開始就向右走7步是一樣的。

to then move

現在，讓我們來看一下向量加法在數字上看起來是怎樣的。

parallel to this new z-axis.

第一個向量的坐標是﹙1, 2﹚，而第二個的坐標是﹙3, -1﹚。

Every triplet of numbers gives you one unique vector in space, and

如果你用這種箭頭到箭尾的方法取向量的和，

every vector in space gives you exactly one triplet of numbers.

你可以想像從原點到第二個箭頭的四步的路徑：

So back to vector addition, and multiplication by numbers.

向右1步，向上2步，向右3步，向下1步。

After all, every topic in linear algebra

重新安排一下這些步驟，

is going to center around these two operations.

使你先做所有向水平的動作，

Luckily, each one is pretty straightforward to define.

然後所有垂直方向的動作，

Let's say we have two vectors, one pointing up, and a little to the right, and the other

你可以這樣說：「先向右動﹙1+3﹚，然後向上﹙2+﹙-1﹚﹚」

one

這樣新的向量就有坐標[1+3, 2+﹙-1﹚]T。

pointing right, and down a bit.

一般來說，在這個數字表格概念裏的向量加法，

To add these two vectors, move the second one so that its tail sits

就是對上它們的項，並把各個加起來。

at the tip of the first one; then if you draw a new vector from the tail of the first one

向量的另一個基本運算是乘以一個數字，

to where

而這最好就先看幾個例子來理解。

the tip of the second one now sits, that new vector is their sum.

如果你拿數字2，把它乘以一個向量，

This definition of addition, by the way, is pretty much the only time in linear algebra

它的意思是拉伸那個向量使它變為2倍長。

where we let

或如果你把那個向量乘上1/3，

vectors stray away from the origin.

它的意思是你把它的長度壓縮到原來長度的1/3。

Now why is this a reasonable thing to do?—Why this definition of addition and not some other

如果你乘以一個負數，像-1.8，

one?

那麽這個向量先翻一個方向，然後拉伸為1.8倍。

Well the way I like to think about it is that each vector represents a certain movement—a

這個拉伸或者壓縮，有時使向量翻轉方向的過程叫做"scaling" ﹙純量乘法﹚

step with

而向量乘上的數字像2或者1/3或者-1.8稱為"scalar" ﹙純量﹚

a certain distance and direction in space.

事實上，在整個的線性代數中，數字所做幾乎都是純量乘法，

If you take a step along the first vector,

因此，「純量」這個字和「數字」幾乎可以互換。

then take a step in the direction and distance described by the second vector, the overall

數字上來說，一個向量乘以一個係數，比方說2，

effect is

相當於它的每一個構件乘以那個係數––2。

just the same as if you moved along the sum of those two vectors to start with.

在向量作爲一些列表數字的概念中，對一個給定的向量乘以一個純量的意思，

You could think about this as an extension of how we think about adding numbers on a

是對每一個構件都乘上那個純量。

number line.

在接下來的影片中，你們將看到在我說線性代數課題自然地圍著這兩個基本運算轉動：

One way that we teach kids to think about this, say with 2+5, is to think of moving

向量和，與純量乘法；

2 steps to the

而我將在最終的影片中講更多關於，為什麽數學家只考慮這兩種運算，

right, followed by another 5 steps to the right.

並將它們抽象獨立出來，使選擇甚麼來代表向量都無關。

The overall effect is the same as if you just took

實際上，你怎樣看待向量都無所謂。

7 steps to the right.

作爲在空間裏的箭頭––就像我建議的––而碰巧有著一個很好的數字表達方式，

In fact, let's see how vector addition looks numerically.

或者根本是來說作爲列表的數字，而碰巧有著一個很好的幾何上的解釋。

The first vector

線性代數的實用性不是在於任何一個獨立觀點，

here has coordinates (1,2), and the second one has coordinates (3,-1).

而是在它們之間的來回轉換，

When you take the vector sum

它提供了一個很好的方法來把大量數字表格概念化、可視化。

using this tip-to-tail method, you can think of a four-step path from the origin to the

使數據中的模式變得清晰，

tip of the

並對某些運作的效果給出一種全面的看法。

second vector: "walk 1 to the right, then 2 up, then 3 to the right, then 1 down."

另一方面，它提供物理學家和電腦圖像工程師一種語言，

Re-organising

讓他們通過電腦能處理的數字來描述井操縱空間。

these steps so that you first do all of the rightward motion, then do all of the vertical

例如，在我制作數學動畫時，

motion,

我首先思考實際在空間裏所發生的，

you can read it as saying, "first move 1+3 to the right, then move 2+(-1) up," so the

然後在電腦上用數字來代表這些變化，

new vector has

從而計算出在屏幕上哪些地方放上像素點，

coordinates 1+3 and 2+(-1).

完成這些通常需要對線性代數的瞭如指掌。

In general, vector addition in this list-of-numbers conception looks

你現在對向量有一些基本了解，而在下一個影片裏，

like matching up their terms, and adding each one together.

我將圍繞向量深入討論到一些像"span" ﹙線性生成空間﹚，"basis" ﹙基﹚和 "linear dependence" ﹙線性相關﹚那些簡明的概念。

The other fundamental vector operation is multiplication by a number.

到時再見！

Now this is best understood

just by looking at a few examples.

If you take the number 2, and multiply it by a given vector, it

means you stretch out that vector so that it's 2 times as long as when you started.

If you multiply

that vector by, say, 1/3, it means you squish it down so that it's 1/3 of the original length.

When you multiply it by a negative number, like -1.8, then the vector first gets flipped

around,

then stretched out by that factor of 1.8.

This process of stretching or squishing or sometimes reversing the direction of a vector

is called "scaling",

and whenever you catch a number like 2 or 1/3 or -1.8 acting like this—scaling some

vector—you call it a "scalar".

In fact, throughout linear algebra, one of the main things that

numbers do is scale vectors, so it's common to use the word "scalar" pretty much interchangeably

with the word "number".

Numerically, stretching out a vector by a factor of, say, 2, corresponds to

multiplying each of its components by that factor, 2, 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.

You'll see in the following videos what I mean when I say that linear algebra topics

tend to revolve

around these two fundamental operations: vector addition, and scalar multiplication; 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.

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, or fundamentally as lists

of numbers

that happen to have a nice geometric interpretation.

The usefulness of linear algebra has less to do with

either one of these views than it does with the ability to translate back and forth between

them.

It gives the data analyst a nice way to conceptualise many lists of numbers in a visual way,

which can seriously clarify patterns in data, and give a global view of what certain operations

do,

and on the flip side, it gives people like physicists and computer graphics programmers

a language

to describe space and the manipulation of space using numbers that can be crunched and

run through a computer.

When I do math-y animations, for example, I start by thinking about what's actually

going on in

space, and then get the computer to represent things numerically, thereby figuring out where

to

place the pixels on the screen, and doing that usually relies on a lot of linear algebra

understanding.

So there are your vector basics, and in the next video I'll start getting into some pretty

neat

concepts surrounding vectors, like span, bases, and linear dependence.

See you then!

Captioned by Navjivan Pal