线性组合、张成空间与基向量 | 线性代数的本质，第二章：英语精读讲义

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

在上个视频中，除了向量加法和标量乘法之外，我还介绍了向量坐标，

where there's this back and forth between, 習語

其中存在着一种来回转换的关系，

for example,

例如，

pairs of numbers and two-dimensional vectors.

在数字对和二维向量之间。

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

现在，我想你们中很多人对向量坐标已经很熟悉了，

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

但还有另一种有趣的方式来思考这些坐标，

which is pretty central to linear algebra.

这在线性代数中处于非常核心的地位。

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

当你有一对用来描述向量的数字时，

like 3 -2,

比如 3 和 -2，

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

我希望你把每个坐标都看作一个标量，

meaning think about how each one stretches or squishes vectors.

意思是想想它们各自是如何拉伸或压缩向量的。

In the XY coordinate system,

在 XY 坐标系中，

there are two very special vectors.

有两个非常特殊的向量。

The one pointing to the right with length one,

一个指向右侧，长度为 1，

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

通常被称为 i-hat，或者 X 方向上的单位向量，

and the one pointing straight up with length one,

另一个径直向上，长度为 1，

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

通常被称为 j-hat，或者 Y 方向上的单位向量。

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

现在，把我们向量的 X 坐标看作是一个缩放 i-hat 的标量，

stretching it by a factor of 3, 短語搭配

将其拉伸为原来的 3 倍，

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

并把 Y 坐标看作是一个缩放 j-hat 的标量，

flipping it and stretching it by a factor of 2.

将其翻转并拉伸为原来的 2 倍。

In this sense,

从这个意义上说，

the vector that these coordinates describe

这些坐标所描述的向量

is the sum of two scaled vectors.

就是两个经过缩放的向量之和。

That's a surprisingly important concept,

这是一个出人意料的重要概念，

this idea of adding together two scaled vectors.

即将两个缩放后的向量相加的想法。

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

顺便说一下，i-hat 和 j-hat 这两个向量

by the way.

有一个特殊的名字。

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

它们一起被称为坐标系的“基”。

What this means basically

基本上，这意味着

is that when you think about coordinates as scalars,

当你把坐标看作标量时，

the basis vectors are what those scalars actually,

基向量就是这些标量实际去，

you know, scale.

你知道，去缩放的对象。

There's also a more technical definition,

这里还有一个更专业的定义，

but I'll get to that later.

但我稍后会讲到。

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

通过用这两个特殊的基向量来构建我们的坐标系，

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

引出了一个非常有趣且微妙的观点。

We could have chosen different basis vectors 語法

我们完全可以选择不同的基向量，

and gotten a completely reasonable new coordinate system.

从而得到一个完全合理的新坐标系。

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

例如，拿一个指向右上方某处的向量，

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

再加上另一个以某种方式指向右下方的向量。

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

花点时间想一想你可以得到的所有不同向量，

by choosing two scalars,

通过选择两个标量，

using each one to scale one of the vectors,

分别用它们来缩放这两个向量中的一个，

then adding together what you get.

然后把得到的结果加起来。

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

通过改变标量的选择，你可以得到哪些二维向量呢？

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

答案是你可以得到每一个可能的二维向量，

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

我觉得思考一下为什么会这样，是个不错的谜题。

A new pair of basis vectors like this

像这样的一对新基向量，

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

仍然为我们提供了一种有效的方式，在数字对

and two-dimensional vectors,

和二维向量之间来回转换，

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

但这种对应关系肯定不同于你使用

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

更标准的 i-hat 和 j-hat 基底所得到的对应关系。

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

这一点我会在以后进行更详细的探讨，

describing the exact relationship between different coordinate systems,

描述不同坐标系之间的确切关系，

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

但就目前而言，我只希望你们能体会到这样一个事实：

any time we describe vectors numerically,

任何时候我们用数字来描述向量，

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

都依赖于我们隐含选择了什么样的基向量。

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

那么，任何时候你像这样缩放两个向量并把它们相加，

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

就被称为这两个向量的“线性组合”。

Where does this word linear come from?

“线性”这个词从何而来？

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

为什么它会和线扯上关系？

Well, this isn't the etymology,

嗯，这不是词源学上的解释，

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

但我喜欢这样去理解：

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

如果你固定其中一个标量，让另一个标量自由改变其数值，

the tip of the resulting vector draws a straight line.

那么合成向量的尖端就会画出一条直线。

Now, if you let both scalars range freely 語法

现在，如果你让两个标量都自由变化，

and consider every possible vector that you can get,

考虑到你能得到的每一个可能的向量，

there are two things that can happen.

可能会发生两件事。

For most pairs of vectors,

对于大多数的向量对，

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

你将能够到达平面上的每一个可能的点。

Every two-dimensional vector is within your grasp. 習語

每一个二维向量都在你的掌控之中。

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

然而，在运气不好的情况下，你最初的两个向量碰巧共线了，

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

那么合成向量的尖端就仅仅被限制在这条穿过原点的直线上。

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

实际上，严格来说，还有第三种可能性。

Both your vectors could be zero,

你的两个向量都可能是零，

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

在这种情况下，你就只能困在原点。

Here's some more terminology.

下面再介绍一些术语。

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

通过给定的一对向量的线性组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

被称为这两个向量的“张成”。

So, restating what we just saw in this lingo,

所以，用这个行话重述我们刚才看到的现象，

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

大多数二维向量对的张成是整个二维空间的所有向量，

but when they line up,

但当它们共线时，

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

它们的张成就是尖端落在某条特定直线上的所有向量。

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

还记得我曾说过，线性代数是围绕向量加法和标量乘法展开的吗？

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

那么，两个向量的张成基本上就是一种提问方式，即

what are all the possible vectors you can reach

你所能到达的所有可能向量是什么

using only these two fundamental operations,

仅仅使用这两种基本运算：

vector addition and scalar multiplication?

向量加法和标量乘法？

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

现在正好适合谈谈人们通常如何把向量看作点。

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

想象整整一组向量落在一条直线上会显得非常拥挤，

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

想象所有二维向量一下子填满整个平面就更拥挤了。

So, when dealing with collections of vectors like this,

所以，在处理像这样的一组向量时，

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

通常只用空间中的一个点来表示每一个向量。

The point at the tip of that vector,

也就是那个向量尖端所在位置的点，

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

当然，像往常一样，我希望你想象那个向量的尾部在原点。

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

这样一来，如果你想思考尖端在某条特定直线上的每一个可能的向量，

just think about the line itself.

只要思考那条直线本身就行了。

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

同样地，要同时思考所有可能的二维向量，

conceptualize each one as the point where its tip sits.

可以把每一个向量概念化为其尖端所在的那个点。

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

这样，实际上你所思考的就是二维空间本身那张无限的平面，

leaving the arrows out of it. 短語動詞

而把箭头抛开不管。

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

一般来说，如果你在单独思考一个向量，

think of it as an arrow.

就把它看作一个箭头。

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

而如果你在处理一组向量，

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

把它们全看作点会更方便。

So, for our span example,

那么，对于我们关于张成的例子，

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

大多数向量对的张成最终就是整个无限的二维平面，

but if they line up,

但如果它们共线，

their span is just a line.

它们的张成就是一条线。

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

如果我们开始思考三维空间中的向量，“张成”的概念会变得有趣得多。

For example,

例如，

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

如果你在三维空间中选取两个指向不同方向的向量，

what does it mean to take their span?

它们的张成意味着什么？

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

嗯，它们的张成就是这两个向量所有可能的线性组合的集合，

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

意思是，通过某种方式缩放它们两个，

and then adding them together.

然后把它们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, 習語

你可以想象一下转动两个不同的旋钮，来改变定义线性组合的这两个标量，

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

把缩放后的向量相加，并跟着合成向量的尖端走。

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

那个尖端会描绘出某种穿过三维空间原点的平面。

This flat sheet is the span of the two vectors,

这个平面就是这两个向量的张成，

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

或者更准确地说，尖端落在那个平面上的所有可能向量的集合就是这两个向量的张成。

Isn't that a beautiful mental image?

这是一个美丽的心理图像，不是吗？

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

那么，如果我们加上第三个向量，并考虑这三个家伙的张成，会发生什么呢？

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

三个向量的线性组合的定义与两个向量的定义非常相似。

You'll choose three different scalars,

你将选择三个不同的标量，

scale each of those vectors,

缩放那几个向量中的每一个，

and then add them all together.

然后把它们全加起来。

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

同样，这些向量的张成就是所有可能线性组合的集合。

Two different things could happen here.

这里可能会发生两种不同的情况。

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

如果你的第三个向量碰巧落在前两个向量的张成平面上，

then the span doesn't change.

那么张成不会改变。

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

你可以说是被困在同一个平面上了。

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

换句话说，在线性组合中加入第三个向量的缩放版本

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

并不能真正让你接触到任何新的向量。

But if you just randomly choose a third vector,

但是如果你只是随机选择一个第三向量，

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

它几乎肯定不会落在前两个向量的张成上。

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

那么，既然它指向一个单独的方向，

it unlocks access to every possible three-dimensional vector. 習語

它就解锁了通往每一个可能的三维向量的通道。

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

我喜欢的一种理解方式是，当你缩放那个新的第三个向量时，

it moves around that span sheet of the first two,

它会在前两个向量张成的那张平面上四处移动，

sweeping it through all of space.

让其扫过整个空间。

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 習語

另一种理解方式是，你充分利用了任你支配的三个可自由改变的标量

to access the full three dimensions of space.

来访问空间的完整三个维度。

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

现在，在第三个向量已经位于前两个向量的张成平面的情况下，

or the case where two vectors happen to line up,

或者是两个向量碰巧共线的情况下，

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

我们需要一些术语来描述这样一个事实：这些向量中至少有一个是多余的，

not adding anything to our span.

并没有给我们的张成增加任何东西。

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

每当这种情况发生，也就是当你有多个向量，且移除其中一个不会缩小张成时，

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

相关的术语是说它们“线性相关”。

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

另一种表达方式是说其中一个向量可以表示为其他向量的线性组合，

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

因为它已经处于其他向量的张成之中。

On the other hand, 表達脈絡

另一方面，

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

如果每一个向量确实都为张成增加了一个新的维度，

they're said to be linearly independent. 用詞

它们就被称为“线性无关”。

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

那么，有了这些术语，希望脑海中也有了与之匹配的良好画面，

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

让我们在结束前留下一个谜题。

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

一个空间的“基”的专业定义是，一组张成该空间的线性无关的向量。

Now, given how I described a basis earlier,

现在，考虑到我之前是如何描述基的，

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

并考虑到你目前对“张成”和“线性无关”这两个词的理解，

think about why this definition would make sense.

想一想为什么这个定义是有道理的。

In the next video,

在下个视频中，

I'll get into matrices in transforming space.

我将探讨矩阵与空间变换。

See you then.

到时见。