線性組合、生成空間與基底向量 | 線性代數的本質，第二章：英語精讀講義

In the last video, along with the ideas of vector addition and scalar multiplication, I described vector coordinates

在上部影片中，除了向量加法和純量乘法的概念外，我還介紹了向量座標，

where there's this back and forth between, 慣用語

其中存在著一種來回轉換的關係，

for example,

例如，

pairs of numbers and two-dimensional vectors.

在數字對和二維向量之間。

Now, I imagine the vector coordinates were already familiar to a lot of you,

現在，我想你們之中很多人對向量座標已經很熟悉了，

but there's another kind of interesting way to think about these coordinates, 表達脈絡

但還有另一種有趣的方式來思考這些座標，

which is pretty central to linear algebra.

這在線性代數中處於非常核心的地位。

When you have a pair of numbers that's meant to describe a vector, 文法

當你有一對用來描述向量的數字時，

like 3 -2,

例如 3 和 -2，

I want you to think about each coordinate as a scalar, 文法

我希望你把每個座標都看作一個純量，

meaning think about how each one stretches or squishes vectors.

意思是想想它們各自是如何拉伸或壓縮向量的。

In the XY coordinate system,

在 XY 座標系中，

there are two very special vectors.

有兩個非常特殊的向量。

The one pointing to the right with length one,

一個指向右側，長度為 1，

commonly called i-hat or the unit vector in the X direction,

通常被稱為 i-hat，或者 X 方向上的單位向量，

and the one pointing straight up with length one,

另一個直指向上，長度為 1，

commonly called j-hat, or the unit vector in the Y direction.

通常被稱為 j-hat，或者 Y 方向上的單位向量。

Now, think of the X coordinate of our vector as a scalar that scales i-hat,

現在，把我們向量的 X 座標看作是一個縮放 i-hat 的純量，

stretching it by a factor of 3, 片語搭配

將其拉伸為原來的 3 倍，

and the Y coordinate as a scalar that scales j-hat,

並把 Y 座標看作是一個縮放 j-hat 的純量，

flipping it and stretching it by a factor of 2.

將其翻轉並拉伸為原來的 2 倍。

In this sense,

從這個意義上說，

the vector that these coordinates describe

這些座標所描述的向量

is the sum of two scaled vectors.

就是兩個經過縮放的向量之和。

That's a surprisingly important concept,

這是一個出乎意料的重要概念，

this idea of adding together two scaled vectors.

即將兩個縮放後的向量相加的想法。

Those two vectors i-hat and j-hat have a special name,

順帶一提，i-hat 和 j-hat 這兩個向量

by the way.

有一個特殊的名字。

Together, they're called the basis of a coordinate system.

它們一起被稱為座標系的「基底」。

What this means basically

這基本上代表著

is that when you think about coordinates as scalars,

當你把座標看作純量時，

the basis vectors are what those scalars actually,

基底向量就是這些純量實際去，

you know, scale.

你知道，去縮放的對象。

There's also a more technical definition,

這裡還有一個更專業的定義，

but I'll get to that later.

但我稍後會講到。

By framing our coordinate system in terms of these two special basis vectors, 片語搭配

透過用這兩個特殊的基底向量來建構我們的座標系，

it raises a pretty interesting and subtle point. 片語搭配

帶出了一個非常有趣且微妙的觀點。

We could have chosen different basis vectors 文法

我們完全可以選擇不同的基底向量，

and gotten a completely reasonable new coordinate system.

從而得到一個完全合理的新座標系。

For example, take some vector pointing up and to the right,

例如，拿一個指向右上方某處的向量，

along with some other vector pointing down and to the right in some way.

再加上另一個以某種方式指向右下方的向量。

Take a moment to think about all the different vectors that you can get

花點時間想一想你可以得到的所有不同向量，

by choosing two scalars,

透過選擇兩個純量，

using each one to scale one of the vectors,

分別用它們來縮放這兩個向量中的一個，

then adding together what you get.

然後把得到的結果加起來。

Which two-dimensional vectors can you reach by altering the choices of scalars?

透過改變純量的選擇，你可以得到哪些二維向量呢？

The answer is that you can reach every possible two-dimensional vector,

答案是你可以得到每一個可能的二維向量，

and I think it's a good puzzle to contemplate why.

我覺得思考一下為什麼會這樣，是個不錯的謎題。

A new pair of basis vectors like this

像這樣的一對新基底向量，

still gives us a valid way to go back and forth between pairs of numbers

仍然為我們提供了一種有效的方式，在數字對

and two-dimensional vectors,

和二維向量之間來回轉換，

but the association is definitely different from the one that you get

但這種對應關係肯定不同於你使用

using the more standard basis of i-hat and j-hat.

更標準的 i-hat 和 j-hat 基底所得到的對應關係。

This is something I'll go into much more detail on later, 片語動詞

這一點我會在以後進行更詳細的探討，

describing the exact relationship between different coordinate systems,

描述不同座標系之間的確切關係，

but for right now, I just want you to appreciate the fact that

但就目前而言，我只希望你們能體會到這樣一個事實：

any time we describe vectors numerically,

任何時候我們用數字來描述向量，

it depends on an implicit choice of what basis vectors we're using. 片語搭配

都取決於我們隱含選擇了什麼樣的基底向量。

So, any time that you're scaling two vectors and adding them like this,

那麼，任何時候你像這樣縮放兩個向量並把它們相加，

it's called a linear combination of those two vectors. 用詞

就被稱為這兩個向量的「線性組合」。

Where does this word linear come from?

「線性」這個詞從何而來？

Why does this have anything to do with lines? 片語搭配

為什麼它會和線（lines）扯上關係？

Well, this isn't the etymology,

嗯，這不是詞源學上的解釋，

but one way I like to think about it is that 表達脈絡

但我喜歡這樣去理解：

if you fix one of those scalars and let the other one change its value freely,

如果你固定其中一個純量，讓另一個純量自由改變其數值，

the tip of the resulting vector draws a straight line.

那麼合成向量的尖端就會畫出一條直線。

Now, if you let both scalars range freely 文法

現在，如果你讓兩個純量都自由變化，

and consider every possible vector that you can get,

考慮到你能得到的每一個可能的向量，

there are two things that can happen.

可能會發生兩件事。

For most pairs of vectors,

對於大多數的向量對，

you'll be able to reach every possible point in the plane.

你將能夠到達平面上的每一個可能的點。

Every two-dimensional vector is within your grasp. 慣用語

每一個二維向量都在你的掌控之中。

However, in the unlucky case where your two original vectors happen to line up, 片語動詞

然而，在運氣不好的情況下，你最初的兩個向量碰巧共線了，

the tip of the resulting vector is limited to just this single line passing through the origin.

那麼合成向量的尖端就僅僅被限制在這條穿過原點的直線上。

Actually, technically, there's a third possibility, too.

實際上，嚴格來說，還有第三種可能性。

Both your vectors could be zero,

你的兩個向量都可能是零，

in which case you'd just be stuck at the origin. 文法

在這種情況下，你就只能困在原點。

Here's some more terminology.

下面再介紹一些術語。

The set of all possible vectors that you can reach with a linear combination of a given pair of vectors

透過給定的一對向量的線性組合所能得到的所有可能向量的集合，

is called the span of those two vectors. 用詞

被稱為這兩個向量的「生成空間」。

So, restating what we just saw in this lingo,

所以，用這個行話重述我們剛才看到的現象，

the span of most pairs of 2D vectors is all vectors of 2D space,

大多數二維向量對的生成空間是整個二維空間的所有向量，

but when they line up,

但當它們共線時，

their span is all vectors whose tip sit on a certain line.

它們的生成空間就是尖端落在某條特定直線上的所有向量。

Remember how I said that linear algebra revolves around vector addition and scalar multiplication? 慣用語

還記得我曾說過，線性代數是圍繞向量加法和純量乘法展開的嗎？

Well, the span of two vectors is basically a way of asking,

那麼，兩個向量的生成空間基本上就是一種提問方式，即

what are all the possible vectors you can reach

你所能到達的所有可能向量是什麼

using only these two fundamental operations,

僅僅使用這兩種基本運算：

vector addition and scalar multiplication?

向量加法和純量乘法？

This is a good time to talk about how people commonly think about vectors as points.

現在正好適合談談人們通常如何把向量看作點。

It gets really crowded to think about a whole collection of vectors sitting on a line,

想像整整一組向量落在一條直線上會顯得非常擁擠，

and more crowded still to think about all two-dimensional vectors all at once filling up the plane.

想像所有二維向量一下子填滿整個平面就更擁擠了。

So, when dealing with collections of vectors like this,

所以，在處理像這樣的一組向量時，

it's common to represent each one with just a point in space.

通常只用空間中的一個點來表示每一個向量。

The point at the tip of that vector,

也就是那個向量尖端所在位置的點，

where, as usual, I want you thinking about that vector with its tail on the origin.

當然，像往常一樣，我希望你想像那個向量的尾部位於原點。

That way, if you want to think about every possible vector whose tip sits on a certain line,

這樣一來，如果你想思考尖端在某條特定直線上的每一個可能的向量，

just think about the line itself.

只要思考那條直線本身就行了。

Likewise, to think about all possible two-dimensional vectors all at once,

同樣地，要同時思考所有可能的二維向量，

conceptualize each one as the point where its tip sits.

可以把每一個向量概念化為其尖端所在的那個點。

So, in effect, what you'll be thinking about is the infinite flat sheet of two-dimensional space itself,

這樣，實際上你所思考的就是二維空間本身那張無限的平面，

leaving the arrows out of it. 片語動詞

而把箭頭拋開不管。

In general, if you're thinking about a vector on its own, 片語搭配

一般來說，如果你在單獨思考一個向量，

think of it as an arrow.

就把它看作一個箭頭。

And if you're dealing with a collection of vectors,

而如果你在處理一組向量，

it's convenient to think of them all as points.

把它們全看作點會更方便。

So, for our span example,

那麼，對於我們關於生成空間的例子，

the span of most pairs of vectors ends up being the entire infinite sheet of two-dimensional space, 片語動詞

大多數向量對的生成空間最終就是整個無限的二維平面，

but if they line up,

但如果它們共線，

their span is just a line.

它們的生成空間就是一條線。

The idea of span gets a lot more interesting if we start thinking about vectors in three-dimensional space.

如果我們開始思考三維空間中的向量，「生成空間」的概念會變得有趣得多。

For example,

例如，

if you take two vectors in 3D space that are not pointing in the same direction,

如果你在三維空間中選取兩個指向不同方向的向量，

what does it mean to take their span?

它們的生成空間意味著什麼？

Well, their span is the collection of all possible linear combinations of those two vectors,

嗯，它們的生成空間就是這兩個向量所有可能的線性組合的集合，

meaning all possible vectors you get by scaling each of the two of them in some way

意思是，透過某種方式縮放它們兩個，

and then adding them together.

然後把它們加在一起所能得到的所有可能的向量。

You can kind of imagine turning two different knobs to change the two scalars defining the linear combination, 慣用語

你可以想像一下轉動兩個不同的旋鈕，來改變定義線性組合的這兩個純量，

adding the scaled vectors and following the tip of the resulting vector.

把縮放後的向量相加，並跟著合成向量的尖端走。

That tip will trace out some kind of flat sheet cutting through the origin of three-dimensional space.

那個尖端會描繪出某種穿過三維空間原點的平面。

This flat sheet is the span of the two vectors,

這個平面就是這兩個向量的生成空間，

or more precisely, the set of all possible vectors whose tips sit on that flat sheet is the span of your two vectors. 表達脈絡

或者更準確地說，尖端落在那個平面上的所有可能向量的集合就是這兩個向量的生成空間。

Isn't that a beautiful mental image?

這是一個美麗的心理圖像，不是嗎？

So, what happens if we add a third vector and consider the span of all three of those guys?

那麼，如果我們加上第三個向量，並考慮這三個傢伙的生成空間，會發生什麼事呢？

A linear combination of three vectors is defined pretty much the same way as it is for two.

三個向量的線性組合的定義與兩個向量的定義非常相似。

You'll choose three different scalars,

你將選擇三個不同的純量，

scale each of those vectors,

縮放那幾個向量中的每一個，

and then add them all together.

然後把它們全加起來。

And again, the span of these vectors is the set of all possible linear combinations.

同樣，這些向量的生成空間就是所有可能線性組合的集合。

Two different things could happen here.

這裡可能會發生兩種不同的情況。

If your third vector happens to be sitting on the span of the first two,

如果你的第三個向量碰巧落在前兩個向量的生成平面上，

then the span doesn't change.

那麼生成空間不會改變。

You're sort of trapped on that same flat sheet.

你可以說是被困在同一個平面上了。

In other words, adding a scaled version of that third vector to the linear combination 表達脈絡

換句話說，在線性組合中加入第三個向量的縮放版本

doesn't really give you access to any new vectors.

並不能真正讓你接觸到任何新的向量。

But if you just randomly choose a third vector,

但是如果你只是隨機選擇一個第三向量，

it's almost certainly not sitting on the span of those first two.

它幾乎肯定不會落在前兩個向量的生成空間上。

Then, since it's pointing in a separate direction,

那麼，既然它指向一個單獨的方向，

it unlocks access to every possible three-dimensional vector. 慣用語

它就解鎖了通往每一個可能的三維向量的通道。

One way I like to think about this is that as you scale that new third vector,

我喜歡的一種理解方式是，當你縮放那個新的第三個向量時，

it moves around that span sheet of the first two,

它會在前兩個向量生成的那張平面上四處移動，

sweeping it through all of space.

讓其掃過整個空間。

Another way to think about it is that you're making full use of the three freely changing scalars that you have at your disposal 慣用語

另一種理解方式是，你充分利用了任你支配的三個可自由改變的純量

to access the full three dimensions of space.

來存取空間的完整三個維度。

Now, in the case where the third vector was already sitting on the span of the first two,

現在，在第三個向量已經位於前兩個向量的生成平面的情況下，

or the case where two vectors happen to line up,

或者是兩個向量碰巧共線的情況下，

we want some terminology to describe the fact that at least one of these vectors is redundant, 用詞

我們需要一些術語來描述這樣一個事實：這些向量中至少有一個是多餘的，

not adding anything to our span.

並沒有給我們的生成空間增加任何東西。

Whenever this happens, where you have multiple vectors and you could remove one without reducing the span,

每當這種情況發生，也就是當你有多個向量，且移除其中一個不會縮小生成空間時，

the relevant terminology is to say that they are linearly dependent. 用詞

相關的術語是說它們「線性相依」。

Another way of phrasing that would be to say that one of the vectors can be expressed as a linear combination of the others, 片語搭配

另一種表達方式是說其中一個向量可以表示為其他向量的線性組合，

since it's already in the span of the others.

因為它已經處於其他向量的生成空間之中。

On the other hand, 表達脈絡

另一方面，

if each vector really does add another dimension to the span, 文法

如果每一個向量確實都為生成空間增加了一個新的維度，

they're said to be linearly independent. 用詞

它們就被稱為「線性獨立」。

So, with all of that terminology and hopefully with some good mental images to go with it,

那麼，有了這些術語，希望腦海中也有了與之匹配的良好畫面，

let me leave you with a puzzle before we go. 片語搭配

讓我們在結束前留下一個謎題。

The technical definition of a basis of a space is a set of linearly independent vectors that span that space. 用詞

一個空間的「基底」的專業定義是，一組生成該空間且線性獨立的向量。

Now, given how I described a basis earlier,

現在，考慮到我之前是如何描述基底的，

and given your current understanding of the words span and linearly independent,

並考慮到你目前對「生成空間」和「線性獨立」這兩個詞的理解，

think about why this definition would make sense.

想一想為什麼這個定義是有道理的。

In the next video,

在下部影片中，

I'll get into matrices in transforming space.

我將探討矩陣與空間變換。

See you then.

到時見。