向量 | 第1章，线性代数的本质：英语精读讲义

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

线性代数最基础的根基和构建模块就是向量。

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

所以，我们非常有必要达成共识，

about what exactly a vector is.

也就是向量到底是什么。

You see, broadly speaking, 表達脈絡

你看，从广义上讲，

there are three distinct but related ideas about vectors,

关于向量有三种不同但相关的观点，

which I'll call the physics student perspective,

我将它们称为物理系学生的视角、

the computer science student perspective,

计算机系学生的视角，

and the mathematician's perspective.

以及数学家的视角。

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

物理系学生的视角认为，向量就是空间中指向某处的箭头。

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

决定一个特定向量的是它的长度和指向的方向。

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

但只要这两个事实保持不变，

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

你可以把它移来移去，它仍然是同一个向量。

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

存在于平面中的向量是二维的，

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

而存在于你我所处的更广阔空间中的向量是三维的。

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

计算机系学生的视角认为，向量是有序的数字列表。

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

例如，假设你在分析房价，

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

而你唯一关心的特征是面积和价格。

You might model each house with a pair of numbers,

你可能会用一对数字来对每栋房子进行建模，

the first indicating square footage and the second indicating price.

第一个表示面积，第二个表示价格。

Notice the order matters here.

注意这里顺序很重要。

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

用行话来说，你就是把房子建模为二维向量，

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

在这种语境下，“向量”几乎只是“列表”的高级说法，

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

而使它成为二维的原因在于该列表的长度为二。

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

另一方面，数学家试图概括这两种观点，

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

基本上是说向量可以是任何具备合理定义的

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

向量相加和向量乘上一个数字的东西，

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

这些操作我会在视频后面详细讲。

The details of this view are rather abstract

这种观点的细节相当抽象，

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

我其实觉得最好在这个系列的最后一个视频之前都先忽略它，

favoring a more concrete setting in the interim. 用詞

在此期间倾向于更具体的设定。

But the reason I bring it up here 短語動詞

但我之所以在这里提出来，

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

是因为它暗示了一个事实：向量加法

and multiplication by numbers will play an important role

和数字乘法的概念将扮演重要角色，

throughout linear algebra.

贯穿整个线性代数。

But before I talk about these operations,

但是在讲这些操作之前，

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

让我们先确定一个具体的想法，以便

when I say the word vector.

当我说到“向量”这个词时在你脑海中浮现。

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

鉴于我在这里主要关注几何，

whenever I introduce a new topic involving vectors,

每当我引入涉及向量的新主题时，

I want you to first think about an arrow,

我希望你首先想到的就是一个箭头，

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

具体来说，想一想在一个坐标系内的箭头，

like the XY plane,

比如 XY 平面，

with its tail sitting at the origin.

它的尾部位于原点。

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

这与物理系学生的视角有些不同，

where vectors can freely sit anywhere they want in space.

在物理中向量可以自由地存在于空间中的任何地方。

In linear algebra,

在线性代数中，

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

几乎所有情况你的向量都将扎根于原点。

Then, once you understand a new concept

然后，一旦你理解了一个新概念，

in the context of arrows in space, 短語搭配

在空间中箭头的语境下，

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

我们会将其转化为数字列表的观点，

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

我们可以通过考虑向量的坐标来实现这一点。

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

那么，虽然我相信你们很多人已经熟悉了这个坐标系，

it's worth walking through explicitly

还是值得详细过一遍，

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

因为这就是所有重要的来回转换发生的地方，

between the two perspectives of linear algebra.

就在线性代数的两种视角之间。

Focusing our attention on two dimensions for the moment,

暂时把注意力集中在二维上，

you have a horizontal line called the x-axis

你有一条水平线称为 x 轴，

and a vertical line called the y-axis.

以及一条垂直线称为 y 轴。

The place where they intersect is called the origin,

它们相交的地方称为原点，

which you should think of as the center of space

你应该把它看作空间的中心

and the root of all vectors.

以及所有向量的根基。

After choosing an arbitrary length to represent one,

在选择了一个代表“1”的任意长度后，

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

你在每个轴上做刻度标记来代表这个距离。

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

当我想传达作为整体的二维空间的概念时，

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

你会看到这在这些视频中经常出现，

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

我会延伸这些刻度标记来制作网格线。

But right now,

但现在，

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

它们其实会有点碍事。

The coordinates of a vector is a pair of numbers

一个向量的坐标是一对数字，

that basically gives instructions for how to get

它基本上给出了关于如何行走的指示，

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

从位于原点的该向量的尾部走向它的尖端。

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

第一个数字告诉你沿 x 轴走多远，

positive numbers indicating rightward motion,

正数表示向右移动，

negative numbers indicating leftward motion,

负数表示向左移动，

and the second number tells you how far to walk

第二个数字告诉你之后需要走多远

parallel to the y-axis after that,

平行于 y 轴，

positive numbers indicating upward motion

正数表示向上移动，

and negative numbers indicating downward motion.

负数表示向下移动。

To distinguish vectors from points,

为了区分向量与点，

the convention is to write this pair of numbers vertically

惯例是将这对数字垂直书写，

with square brackets around them.

并用方括号括起来。

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

每一对数字都能给你提供且仅提供一个向量，

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

而每一个向量也与唯一一对数字相关联。

What about in three dimensions?

那么在三维中呢？

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

嗯，你增加第三个轴，称为 z 轴，

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

它垂直于 x 轴和 y 轴，

and in this case,

在这种情况下，

each vector is associated with an ordered triplet of numbers.

每个向量都与一个有序的三元数组相关联。

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

第一个告诉你沿 x 轴移动多远，

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

第二个告诉你平行于 y 轴移动多远，

and the third one tells you how far to then move

第三个则告诉你随后要移动多远

parallel to this new z-axis.

平行于这个新的 z 轴。

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

每一个三元数组都能在空间中给你提供一个唯一的向量，

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

空间中的每一个向量也能准确地对应一个三元数组。

All right,

好了，

so back to vector addition and multiplication by numbers.

让我们回到向量的加法和数字乘法。

After all, 表達脈絡

毕竟，

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

线性代数中的每个主题都将围绕这两个操作展开。

Luckily, each one's pretty straightforward to define.

幸运的是，每一个的定义都很直接。

Let's say we have two vectors,

假设我们有两个向量，

one pointing up and a little to the right

一个指向上方且略偏右，

and the other one pointing right and down a bit.

另一个指向右侧且略偏下。

To add these two vectors,

要将这两个向量相加，

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

移动第二个向量，使它的尾部位于第一个向量的尖端。

Then, if you draw a new vector

然后，如果你画一个新向量

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

从第一个向量的尾部到第二个向量尖端现在所在的位置，

that new vector is their sum.

那个新向量就是它们的和。

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

顺便说一句，这个加法的定义，

is pretty much the only time in linear algebra

差不多是线性代数中唯一一次

where we let vectors stray away from the origin.

我们让向量偏离原点。

Now,

那么，

why is this a reasonable thing to do?

为什么这样做是合理的呢？

Why this definition of addition and not some other one?

为什么用这种加法定义而不是别的什么呢？

Well,

这个嘛，

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

我喜欢的思考方式是每个向量代表一种特定的运动，

a step with a certain distance and direction in space.

在空间中具有特定距离和方向的一步。

If you take a step along the first vector,

如果你沿着第一个向量迈出一步，

then take a step in the direction and distance

然后朝着该方向和距离迈出一步

described by the second vector,

根据第二个向量所描述的，

the overall effect is just the same as if you moved

整体效果就和你一开始就移动

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

沿着那两个向量的和是一样的。

You could think about this as an extension

你可以把这看作是一种延伸，

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

就像我们看待数轴上的数字相加一样。

One way that we teach kids to think about this,

我们教小孩子思考这个问题的一种方法，

say with 2 + 5,

比如说 2 + 5，

is to think of moving two steps to the right,

是想象向右移动两步，

followed by another five steps to the right.

紧接着再向右移动五步。

The overall effect is the same

整体的效果是一样的，

as if you just took seven steps to the right.

就好像你直接向右移动了七步一样。

In fact,

事实上，

let's see how vector addition looks numerically.

让我们看看向量加法在数值上是什么样的。

The first vector here has coordinates 1 2,

这里的第一个向量坐标是 1 2，

and the second one has coordinates 3 -1.

第二个向量的坐标是 3 -1。

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

当你使用这种首尾相接的方法进行向量相加时，

you can think of a four-step path

你可以想象一条四步路径，

from the origin to the tip of the second vector.

从原点到第二个向量的尖端。

Walk one to the right,

向右走一格，

then two up,

然后向上走两格，

then three to the right,

然后向右走三格，

then one down.

然后向下走一格。

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

重新组织这些步骤，以便你先做所有向右的运动，

then do all the vertical motion,

然后做所有的垂直运动，

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

你可以将其解读为先向右移动 1 + 3，

then move 2 -1 up.

然后向上移动 2 -1。

So,

所以，

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

新向量的坐标是 1 + 3 和 2 + -1。

In general,

一般来说，

vector addition in this list of numbers conception

在这种数字列表概念下的向量加法

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

看起来就像是把它们的各项对应起来并逐个相加。

The other fundamental vector operation

另一个基本的向量操作

is multiplication by a number.

就是乘以一个数字。

Now,

这个嘛，

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

只需看几个例子就能最好地理解。

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

如果你取数字二并乘以给定的一个向量，

it means you stretch out that vector 短語動詞

这意味着你拉伸那个向量，

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

使得它的长度是你开始时的两倍。

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

如果你将那个向量乘以，比如说，1/3，

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

这意味着你把它压缩，使它变成原来长度的 1/3。

When you multiply it by a negative number,

当你乘以一个负数时，

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

比如 -1.8，那么向量会先反转方向，

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

然后按 1.8 的系数拉伸。

This process of stretching or squishing

这个拉伸、压缩

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

或有时反转向量方向的过程称为缩放。

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

而每当你看到一个数字，比如二、1/3 或 -1.8，

acting like this, scaling some vector,

像这样起作用，缩放某个向量时，

you call it a scalar.

你称它为标量。

In fact, throughout linear algebra,

事实上，在整个线性代数中，

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

数字所做的主要事情之一就是缩放向量。

So,

所以，

it's common to use the word scalar

使用“标量”这个词很常见，

pretty much interchangeably with the word number. 短語搭配

几乎可以和“数字”这个词互换使用。

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

在数值上，将一个向量拉伸一个系数，比如说两倍，

corresponds with multiplying each of its components by that factor,

对应于将其每个分量乘以那个系数，

two.

也就是二。

So,

所以，

in the conception of vectors as lists of numbers,

在将向量视为数字列表的概念中，

multiplying a given vector by a scalar

将给定向量乘以一个标量

means multiplying each one of those components by that scalar.

意味着将其中每一个分量都乘以该标量。

You'll see in the following videos what I mean

你会在接下来的视频中明白我的意思，

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

当我说线性代数主题往往围绕着

around these two fundamental operations,

这两个基本操作，

vector addition and scalar multiplication.

向量加法和标量乘法。

And I'll talk more in the last video

而且我会在最后一个视频中更多地谈论

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

为什么以及数学家如何只考虑这些操作，

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

独立于并且抽象于你选择表示向量的任何方式。

In truth,

事实上，

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

你是认为向量本质上是空间中的箭头

like I'm suggesting you do,

就像我建议你做的那样，

that happen to have a nice numerical representation, 語法

碰巧有一个很好的数字表示，

or fundamentally as lists of numbers

还是认为它本质上是数字列表

that happen to have a nice geometric interpretation.

碰巧有一个很好的几何解释，都不重要。

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

线性代数的实用性与这两种观点本身的关联较少，

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

而更多地在于能够在它们之间来回转换的能力。

It gives the data analyst a nice way to conceptualize

它为数据分析师提供了一种很好的方式来进行概念化，

many lists of numbers in a visual way,

以视觉方式呈现大量的数字列表，

which can seriously clarify patterns in data

这能够极大地阐明数据中的模式，

and give a global view of what certain operations do.

并提供某些操作所产生的全局视野。

And on the flip side, 表達脈絡

另一方面，

it gives people like physicists and computer graphics programmers

它赋予了物理学家和计算机图形程序员等人

a language to describe space

一种描述空间的语言，

and the manipulation of space

以及操纵空间的能力，

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

使用可以通过计算机处理和运行的数字。

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

例如，当我制作具有数学特色的动画时，

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

我首先会思考空间中实际发生的事情，

and then get the computer to represent things numerically,

然后让计算机以数字方式表示事物，

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

从而弄清楚在屏幕上的什么位置放置像素。

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

而要做到这一点，通常依赖于大量的线性代数知识。

So,

那么，

there are your vector basics,

这些就是你的向量基础知识，

and in the next video,

在下一个视频中，

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

我将开始探讨一些围绕向量的相当绝妙的概念，

like span, bases, and linear dependence.

比如张成空间、基底以及线性相关性。

See you then.

到时候见。