向量 | 第1章，線性代數的本質：英語精讀講義

The fundamental root of it all building block for linear algebra is the vector.

線性代數最根本的基礎與建構區塊就是向量。

So, it's worth making sure that we're all on the same page 講解

因此，我們非常有必要達成共識，

about what exactly a vector is.

也就是向量到底是什麼。

You see, broadly speaking, 表達脈絡

你看，廣義來說，

there are three distinct but related ideas about vectors,

關於向量有三種不同但相關的觀點，

which I'll call the physics student perspective,

我將它們稱為物理系學生的視角、

the computer science student perspective,

資訊工程系學生的視角，

and the mathematician's perspective.

以及數學家的視角。

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

物理系學生的視角認為，向量是空間中指向某處的箭頭。

What defines a given vector is its length and the direction it's pointing.

決定一個特定向量的是它的長度與指向的方向。

But as long as those two facts are the same, 文法

但只要這兩個事實保持不變，

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

你可以把它移來移去，它仍然是同一個向量。

Vectors that live in the flat plane are two-dimensional,

存在於平面中的向量是二維的，

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

而存在於你我所處的更廣闊空間中的向量是三維的。

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

資工系學生的視角認為，向量是有序的數字列表。

For example, let's say you were doing some analytics about house prices,

例如，假設你在分析房價，

and the only features you cared about were square footage and price.

而你唯一在乎的特徵是坪數與價格。

You might model each house with a pair of numbers,

你可能會用一對數字來對每棟房子進行建模，

the first indicating square footage and the second indicating price.

第一個表示坪數，第二個表示價格。

Notice the order matters here.

注意這裡順序很重要。

In the lingo, you'd be modeling houses as two-dimensional vectors,

用行話來說，你就是把房子建模為二維向量，

where in this context vector is pretty much just a fancy word for list,

在這種脈絡下，「向量」幾乎只是「列表」的高級說法，

and what makes it two-dimensional is the fact that the length of that list is two.

而使它成為二維的原因在於該列表的長度為二。

The mathematician, on the other hand, seeks to generalize both these views, 表達脈絡

另一方面，數學家試圖概括這兩種觀點，

basically saying that a vector can be anything where there's a sensible notion

基本上是說向量可以是任何具備合理定義的

of adding two vectors and multiplying a vector by a number,

兩個向量相加以及向量乘上一個數字的東西，

operations that I'll talk about later on in this video.

這些運算我會在影片後面詳細談。

The details of this view are rather abstract

這種觀點的細節相當抽象，

and I actually think it's healthy to ignore it until the last video of this series,

我其實覺得最好在這個系列的最後一部影片之前都先忽略它，

favoring a more concrete setting in the interim. 用詞

在此期間傾向於更具體的設定。

But the reason I bring it up here 片語動詞

但我之所以在這裡提出來，

is that it hints at the fact that the ideas of vector addition

是因為它暗示了一個事實：向量加法

and multiplication by numbers will play an important role

與數字乘法的概念將扮演重要角色，

throughout linear algebra.

貫穿整個線性代數。

But before I talk about these operations,

但是在談論這些運算之前，

let's just settle in on a specific thought to have in mind

讓我們先確定一個具體的想法，以便

when I say the word vector.

當我說到「向量」這個詞時在你腦海中浮現。

Given the geometric focus that I'm shooting for here, 片語動詞

鑑於我在這裡主要關注幾何，

whenever I introduce a new topic involving vectors,

每當我引入涉及向量的新主題時，

I want you to first think about an arrow,

我希望你首先想到的就是一個箭頭，

and specifically, think about that arrow inside a coordinate system,

具體來說，想一想在一個座標系內的箭頭，

like the XY plane,

比如 XY 平面，

with its tail sitting at the origin.

它的尾部停在原點。

This is a little bit different from the physics student perspective,

這與物理系學生的視角有些不同，

where vectors can freely sit anywhere they want in space.

在物理中向量可以自由地存在於空間中的任何地方。

In linear algebra,

在線性代數中，

it's almost always the case that your vector will be rooted at the origin.

幾乎所有情況下你的向量都會紮根於原點。

Then, once you understand a new concept

然後，一旦你理解了一個新概念，

in the context of arrows in space, 片語搭配

在空間中箭頭的脈絡下，

we'll translate it over to the list of numbers point of view,

我們會將其轉化為數字列表的觀點，

which we can do by considering the coordinates of the vector.

我們可以透過考慮向量的座標來實現這一點。

Now, while I'm sure that many of you are already familiar with this coordinate system,

那麼，雖然我相信你們很多人已經熟悉了這個座標系，

it's worth walking through explicitly

還是值得詳細走一遍，

since this is where all of the important back and forth happens

因為這就是所有重要的來回轉換發生的地方，

between the two perspectives of linear algebra.

就在線性代數的兩種視角之間。

Focusing our attention on two dimensions for the moment,

暫時把注意力集中在二維上，

you have a horizontal line called the x-axis

你有一條水平線稱為 x 軸，

and a vertical line called the y-axis.

以及一條垂直線稱為 y 軸。

The place where they intersect is called the origin,

它們相交的地方稱為原點，

which you should think of as the center of space

你應該把它看作空間的中心

and the root of all vectors.

以及所有向量的根基。

After choosing an arbitrary length to represent one,

在選擇了一個代表「1」的任意長度後，

you make tick marks on each axis to represent this distance.

你在每個軸上做刻度標記來代表這個距離。

When I want to convey the idea of 2D space as a whole,

當我想傳達作為整體的二維空間的概念時，

which you'll see comes up a lot in these videos, 片語動詞

你會看到這在這些影片中經常出現，

I'll extend these tick marks to make grid lines.

我會延伸這些刻度標記來製作網格線。

But right now,

但現在，

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

它們其實會有點礙事。

The coordinates of a vector is a pair of numbers

一個向量的座標是一對數字，

that basically gives instructions for how to get

它基本上給出了關於如何行走的指示，

from the tail of that vector at the origin to its tip.

從位於原點的該向量的尾部走向它的尖端。

The first number tells you how far to walk along the x-axis,

第一個數字告訴你沿 x 軸走多遠，

positive numbers indicating rightward motion,

正數表示向右移動，

negative numbers indicating leftward motion,

負數表示向左移動，

and the second number tells you how far to walk

第二個數字告訴你之後需要走多遠

parallel to the y-axis after that,

平行於 y 軸，

positive numbers indicating upward motion

正數表示向上移動，

and negative numbers indicating downward motion.

負數表示向下移動。

To distinguish vectors from points,

為了區分向量與點，

the convention is to write this pair of numbers vertically

慣例是將這對數字垂直書寫，

with square brackets around them.

並用中括號括起來。

Every pair of numbers gives you one and only one vector,

每一對數字都能給你提供且僅提供一個向量，

and every vector is associated with one and only one pair of numbers.

而每一個向量也與唯一一對數字相關聯。

What about in three dimensions?

那麼在三維中呢？

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

嗯，你增加第三個軸，稱為 z 軸，

which is perpendicular to both the x and y-axes,

它垂直於 x 軸與 y 軸，

and in this case,

在這種情況下，

each vector is associated with an ordered triplet of numbers.

每個向量都與一個有序的三元數字組相關聯。

The first tells you how far to move along the x-axis,

第一個告訴你沿 x 軸移動多遠，

the second tells you how far to move parallel to the y-axis,

第二個告訴你平行於 y 軸移動多遠，

and the third one tells you how far to then move

第三個則告訴你隨後要移動多遠

parallel to this new z-axis.

平行於這個新的 z 軸。

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

每一個三元數字組都能在空間中給你提供一個唯一的向量，

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

空間中的每一個向量也能準確地對應一個三元數字組。

All right,

好了，

so back to vector addition and multiplication by numbers.

讓我們回到向量加法與數字乘法。

After all, 表達脈絡

畢竟，

every topic in linear algebra is going to center around these two operations.

線性代數中的每個主題都將圍繞這兩個運算展開。

Luckily, each one's pretty straightforward to define.

幸運的是，每一個的定義都很直接。

Let's say we have two vectors,

假設我們有兩個向量，

one pointing up and a little to the right

一個指向上方且略偏右，

and the other one pointing right and down a bit.

另一個指向右側且略偏下。

To add these two vectors,

要將這兩個向量相加，

move the second one so that its tail sits at the tip of the first one.

移動第二個向量，使它的尾部停在第一個向量的尖端。

Then, if you draw a new vector

然後，如果你畫一個新向量

from the tail of the first one to where the tip of the second one now sits,

從第一個向量的尾部到第二個向量尖端現在所在的位置，

that new vector is their sum.

那個新向量就是它們的總和。

This definition of addition, by the way, 表達脈絡

順帶一提，這個加法的定義，

is pretty much the only time in linear algebra

差不多是線性代數中唯一一次

where we let vectors stray away from the origin.

我們讓向量偏離原點。

Now,

那麼，

why is this a reasonable thing to do?

為什麼這樣做是合理的呢？

Why this definition of addition and not some other one?

為什麼用這種加法定義而不是別的什麼呢？

Well,

這個嘛，

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

我喜歡的思考方式是每個向量代表一種特定的運動，

a step with a certain distance and direction in space.

在空間中具有特定距離與方向的一步。

If you take a step along the first vector,

如果你沿著第一個向量邁出一步，

then take a step in the direction and distance

然後朝著該方向與距離邁出一步

described by the second vector,

根據第二個向量所描述的，

the overall effect is just the same as if you moved

整體效果就和你一開始就移動

along the sum of those two vectors to start with. 片語搭配

沿著那兩個向量的總和是一樣的。

You could think about this as an extension

你可以把這看作是一種延伸，

of how we think about adding numbers on a number line.

就像我們看待數線上數字相加一樣。

One way that we teach kids to think about this,

我們教小孩子思考這個問題的一種方法，

say with 2 + 5,

比如說 2 + 5，

is to think of moving two steps to the right,

是想像向右移動兩步，

followed by another five steps to the right.

緊接著再向右移動五步。

The overall effect is the same

整體的效果是一樣的，

as if you just took seven steps to the right.

就好像你直接向右移動了七步一樣。

In fact,

事實上，

let's see how vector addition looks numerically.

讓我們看看向量加法在數值上是什麼樣子。

The first vector here has coordinates 1 2,

這裡的第一個向量座標是 1 2，

and the second one has coordinates 3 -1.

第二個向量的座標是 3 -1。

When you take the vector sum using this tip-to-tail method, 用詞

當你使用這種首尾相接的方法進行向量相加時，

you can think of a four-step path

你可以想像一條四步路徑，

from the origin to the tip of the second vector.

從原點到第二個向量的尖端。

Walk one to the right,

向右走一格，

then two up,

然後向上走兩格，

then three to the right,

然後向右走三格，

then one down.

然後向下走一格。

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

重新組織這些步驟，以便你先做所有向右的運動，

then do all the vertical motion,

然後做所有的垂直運動，

you can read it as saying first move 1 + 3 to the right,

你可以將其解讀為先向右移動 1 + 3，

then move 2 -1 up.

然後向上移動 2 -1。

So,

所以，

the new vector has coordinates 1 + 3 and 2 + -1.

新向量的座標是 1 + 3 和 2 + -1。

In general,

一般來說，

vector addition in this list of numbers conception

在這種數字列表概念下的向量加法

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

看起來就像是把它們的各項對應起來並逐個相加。

The other fundamental vector operation

另一個基本的向量運算

is multiplication by a number.

就是乘以一個數字。

Now,

這個嘛，

this is best understood just by looking at a few examples. 文法

只需看幾個例子就能最好地理解。

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

如果你取數字二並乘以給定的一個向量，

it means you stretch out that vector 片語動詞

這意味著你拉伸那個向量，

so that it's two times as long as when you started. 文法

使得它的長度是你開始時的兩倍。

If you multiply that vector by, say, 1/3,

如果你將那個向量乘以，比如說，1/3，

it means you squish it down so that it's 1/3 the original length. 片語動詞

這意味著你把它壓縮，使它變成原來長度的 1/3。

When you multiply it by a negative number,

當你乘以一個負數時，

like -1.8, then the vector first gets flipped around, 片語動詞

比如 -1.8，那麼向量會先反轉方向，

then stretched out by that factor of 1.8. 片語搭配

然後按 1.8 的倍數拉伸。

This process of stretching or squishing

這個拉伸、壓縮

or sometimes reversing the direction of a vector is called scaling.

或有時反轉向量方向的過程稱為縮放。

And whenever you catch a number, like two or 1/3 or -1.8,

而每當你看到一個數字，比如二、1/3 或 -1.8，

acting like this, scaling some vector,

像這樣起作用，縮放某個向量時，

you call it a scalar.

你稱它為純量。

In fact, throughout linear algebra,

事實上，在整個線性代數中，

one of the main things that numbers do is scale vectors.

數字所做的主要事情之一就是縮放向量。

So,

所以，

it's common to use the word scalar

使用「純量」這個詞很常見，

pretty much interchangeably with the word number. 片語搭配

幾乎可以和「數字」這個詞互換使用。

Numerically, stretching out a vector by a factor of, say, two

在數值上，將一個向量拉伸一個倍數，比如說兩倍，

corresponds with multiplying each of its components by that factor,

對應於將其每個分量乘以那個倍數，

two.

也就是二。

So,

所以，

in the conception of vectors as lists of numbers,

在將向量視為數字列表的概念中，

multiplying a given vector by a scalar

將給定向量乘以一個純量

means multiplying each one of those components by that scalar.

意味著將其中每一個分量都乘以該純量。

You'll see in the following videos what I mean

你會在接下來的影片中明白我的意思，

when I say that linear algebra topics tend to revolve 片語搭配

當我說線性代數主題往往圍繞著

around these two fundamental operations,

這兩個基本運算，

vector addition and scalar multiplication.

向量加法與純量乘法。

And I'll talk more in the last video

而且我會在最後一部影片中更多地談論

about how and why the mathematician thinks only about these operations,

為什麼以及數學家如何只考慮這些運算，

independent and abstracted away from however you choose to represent vectors. 片語動詞

獨立於並且抽象於你選擇表示向量的任何方式。

In truth,

事實上，

it doesn't matter whether you think about vectors as fundamentally being arrows in space, 文法

你是認為向量本質上是空間中的箭頭

like I'm suggesting you do,

就像我建議你做的那樣，

that happen to have a nice numerical representation, 文法

碰巧有一個很好的數字表示，

or fundamentally as lists of numbers

還是認為它本質上是數字列表

that happen to have a nice geometric interpretation.

碰巧有一個很好的幾何解釋，都不重要。

The usefulness of linear algebra has less to do with either one of these views 文法

線性代數的實用性與這兩種觀點本身的關聯較少，

than it does with the ability to translate back and forth between them. 片語搭配

而更多地在於能夠在它們之間來回轉換的能力。

It gives the data analyst a nice way to conceptualize

它為數據分析師提供了一種很好的方式來進行概念化，

many lists of numbers in a visual way,

以視覺方式呈現大量的數字列表，

which can seriously clarify patterns in data

這能夠極大地闡明數據中的模式，

and give a global view of what certain operations do.

並提供某些運算所產生的全域視野。

And on the flip side, 表達脈絡

另一方面，

it gives people like physicists and computer graphics programmers

它賦予了物理學家與電腦圖學程式設計師等人

a language to describe space

一種描述空間的語言，

and the manipulation of space

以及操縱空間的能力，

using numbers that can be crunched and run through a computer. 用詞

使用可以透過電腦處理和運算的數字。

When I do math-y animations, for example, 用詞

例如，當我製作具有數學特色的動畫時，

I start by thinking about what's actually going on in space 慣用語

我首先會思考空間中實際發生的事情，

and then get the computer to represent things numerically,

然後讓電腦以數字方式表示事物，

thereby figuring out where to place the pixels on the screen. 文法

從而弄清楚在螢幕上的什麼位置放置像素。

And doing that usually relies on a lot of linear algebra understanding.

而要做到這一點，通常依賴於大量的線性代數知識。

So,

那麼，

there are your vector basics,

這些就是你的向量基礎知識，

and in the next video,

在下一部影片中，

I'll start getting into some pretty neat concepts surrounding vectors,

我將開始探討一些圍繞向量的相當絕妙的概念，

like span, bases, and linear dependence.

比如生成空間、基底以及線性相依。

See you then.

到時候見。