跟读练习: Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra - 通过视频学习英语口语

选择练习模式

每种模式都会打开全屏练习室。

91 句

如果句子过短或过长，请点击 Edit 进行调整。

In the last video, along with the ideas of vector addition and scalar multiplication, I described vector coordinates, where there's this back and forth between,

for example, pairs of numbers and two -dimensional vectors.

Now, I imagine the vector coordinates were already familiar to a lot of you, but there's another kind of interesting way to think about these coordinates, which is pretty central to linear algebra.

When you have a pair of numbers that's meant to describe a vector, like 3, negative 2, I want you to think about each coordinate as a scalar,

meaning think about how each one stretches or squishes vectors.

In the xy coordinate system, there are two very special vectors, the one pointing to the right with length 1, commonly called i hat, or the unit vector in the x direction,

and the one pointing straight up with length 1 commonly called j hat, or the unit vector in the y direction.

Now, think of the x coordinate of our vector as a scalar that scales i hat, stretching it by a factor of 3.

And the y coordinate as a scalar that scales j hat, flipping it and stretching it by a factor of 2.

In this sense, the vector that these coordinates describe is the sum of two scaled vectors.

That's a surprisingly important concept, this idea of adding together two scaled vectors.

Those two vectors, i -hat and j -hat, have a special name, by the way.

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

What this means, basically, is that when you think about coordinates as scalars, the basis vectors are what those scalars actually, you know, scale.

There's also a more technical definition, but I'll get to that later.

By framing our coordinate system in terms of these two special basis vectors, it raises a pretty interesting and subtle point.

We could have chosen different basis vectors and gotten a completely reasonable new coordinate system.

For example, take some vector pointing up and to the right, along with some other vector pointing down and to the right in some way.

Take a moment to think about all the different vectors that you can get by choosing two scalars, using each one to scale one of the vectors, then adding together what you get.

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

The answer is that you can reach every possible two -dimensional vector, and I think it's a good puzzle to contemplate why.

A new pair of basis vectors like this still gives us a valid way to go back

and forth between pairs of numbers and two -dimensional vectors, but the association is definitely different from the one

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

This is something I'll go into much more detail on later, describing the exact relationship between different coordinate systems,

But for right now, I just want you to appreciate the fact that anytime we describe vectors numerically, it depends on an implicit choice of what basis vectors we're using.

So anytime that you're scaling two vectors and adding them like this, it's called a linear combination of those two vectors.

Where does this word linear come from?

Why does this have anything to do with lines?

Well, this isn't the etymology, but one way I like to think about it is that

if you fix one of those scalars and let the other one change its value freely, the tip of the resulting vector draws a straight line.

Now if you let both scalars range freely, and consider every possible vector that you can get, there are two things that can happen.

For most pairs of vectors, you'll be able to reach every possible point in the plane.

Every two -dimensional vector is within your grasp.

However, in the unlucky case where your two original vectors happen to line up, the tip of the resulting vector is limited to just this single line passing through the origin.

Actually, technically there's a third possibility too.

Both your vectors could be zero, in which case you'd just be stuck at the origin.

Here's some more terminology.

The set of all possible vectors

that you can reach with a linear combination of a given pair of vectors is called the span of those two vectors.

So restating what we just saw in this lingo, the span of most pairs of 2D vectors is all vectors of 2D space, but when they line up,

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

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

Well, the span of two vectors is basically a way of asking, what are all the possible vectors you can reach using only these two fundamental operations,

vector addition and scalar multiplication?

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

It gets really crowded to think about a whole collection of vectors sitting on a line, and more crowded still to think about all two -dimensional vectors all at once, filling up the plane.

So, when dealing with collections of vectors like this, it's common to represent each one with just a point in space.

The point at the tip of that vector where, as usual, I want you thinking about that vector with its tail on the origin.

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

Likewise, to think about all possible two -dimensional vectors all at once, conceptualize each one as the point where its tip sits.

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

In general, if you're thinking about a vector on its own, think of it as an arrow.

And if you're dealing with a collection of vectors, it's convenient to think of them all as points.

So, for our span example, the span of most pairs of vectors ends up being the entire infinite sheet of two -dimensional space

but if they line up, their span is just a line.

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

For example, if you take two vectors in 3D space that are not pointing in the same direction, what does it mean to take their span?

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

meaning all possible vectors you get by scaling each of the two of them in some way and then adding them together.

You can kind of imagine turning two different knobs to change the two scalars defining the linear combination, adding the scaled vectors and following the tip of the resulting vector.

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

This flat sheet is the span of the two vectors, Or, more precisely, the set of all possible vectors whose tips sit on

that flat sheet is the span of your two vectors.

Isn't that a beautiful mental image?

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

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

You'll choose three different scalars, scale each of those vectors, and then add them all together.

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

Two different things could happen here.

If your third vector happens to be sitting on the span of the first two, then the span doesn't change.

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

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

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

But if you just randomly choose a third vector, it's almost certainly not sitting on the span of those first two.

Then, since it's pointing in a separate direction, it unlocks access to every possible three -dimensional vector.

One way I like to think about this is that as you scale that new third vector, it moves around that span sheet of the first two, sweeping it through all of space.

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 to access the full three dimensions of space.

Now, in the case where the third vector was already sitting on the span of the first two, or the case where two vectors happen to line up,

we want some terminology to describe the fact that at least one of these vectors is redundant, not adding anything to our span.

Whenever this happens, where you have multiple vectors and you could remove one without reducing the span, the relevant terminology is to say that they are linearly dependent.

Another way of phrasing that would be to say

that one of the vectors can be expressed as a linear combination of the others, since it's already in the span of the others.

On the other hand, if each vector really does add another dimension to the span, they're said to be linearly independent.

So with all of that terminology, and hopefully with some good mental images to go with it, let me leave you with a puzzle before we go.

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

Now, given how I described a basis earlier, and given your current understanding of the words span

and linearly independent think about why this definition would make sense in the next video I'll get into matrices

and transforming space see you then

下载应用

流利口语所需的一切

AI 发音评分逐句打分

音标练习掌握每个音

词汇学习建立你的词库

词汇游戏边玩边学

扫码

Download on theApp Store Get it onGoogle Play

同频道

FOR YOU

本课的词汇与口语要点

这节 C1 级别的口语课以视频“Linear combinations, span, and basis vectors”为素材。视频中反复出现的词有：vector, span, scalar, coordinate, dimensional。这段视频共有 91 个句子、1667 个单词可供跟读。讲话部分时长为 9:26。说话人语速自然，每分钟约 177 个词，接近日常对话。只有 79% 的单词属于英语最常用的 3,000 词，词汇难度较高。

视频中的重点词汇

视频中最难的 15 个单词，附发音和释义：

单词	发音	释义
vector 名词	/ˈvɛktɚ/	向量, 矢量
scalar 形容词	/ˈskeɪ.lə/	標量 /标量
dimensional 形容词	/daɪˈmɛn.ʃə.nəl/	維的 /维的
terminology 名词	/ˌtɝ.məˈnɑ.lə.d͡ʒi/	術語學 /术语学
multiplication 名词	/ˌmʌltɪplɪˈkeɪʃən/	乘法
algebra 名词	/ˈæl.d͡ʒɪ.bɹə/	代數 /代数
puzzle 名词	/ˈpʌzəl/	謎 /谜, 難題 /难题
dimension 名词	/daɪˈmɛn.ʃən/	方面, 特點 /特点
conceptualize 动词		概念化
etymology 名词	/ˌɛtəˈmɑləd͡ʒi/	詞源 /词源, 語源 /语源
implicit 形容词	/ɪmˈplɪsɪt/	隱含的 /隐含的, 隐含的
revolve 动词	/ɹɪˈvɑlv/	圍繞 /围绕
redundant 形容词	/ɹɪˈdʌn.dənt/	冗雜 /冗杂, 冗餘 /冗余
unlucky 形容词	/ʌnˈlʌki/	不幸, 不走運 /不走运
unlock 动词	/ʌnˈlɑk/	開鎖 /开锁, 解鎖 /解锁

视频中出现的短语动词

单词	释义
end up 动词	到达
fill up 动词	裝滿 /装满
go back 动词	回去, 返回

需要注意的发音

说话人用了 19 次缩略和弱读形式，例如 you're, I'll, you'll。请按听到的简短形式来说。

“sh” 和 “zh” 音: dimensional /daɪˈmɛn.ʃə.nəl/, multiplication /ˌmʌltɪplɪˈkeɪʃən/, combinations /kɑmbɪˈneɪʃənz/, dimension /daɪˈmɛn.ʃən/
长单词——注意重音位置: coordinate /koʊˈɔɹ.dəˌneɪt/, dimensional /daɪˈmɛn.ʃə.nəl/, terminology /ˌtɝ.məˈnɑ.lə.d͡ʒi/, multiplication /ˌmʌltɪplɪˈkeɪʃən/, combinations /kɑmbɪˈneɪʃənz/

中文母语者容易读错的音：

/v/ — 不要读成 /w/，上齿轻触下唇: vector /ˈvɛktɚ/, revolve /ɹɪˈvɑlv/
词尾辅音 — 要读清楚，后面不要加元音: coordinate /koʊˈɔɹ.dəˌneɪt/, combinations /kɑmbɪˈneɪʃənz/, contemplate /ˈkɑn.təmˌpleɪt/, implicit /ɪmˈplɪsɪt/, revolve /ɹɪˈvɑlv/
/r/ 和 /l/ 的区别: freely /ˈfɹili/, algebra /ˈæl.d͡ʒɪ.bɹə/, revolve /ɹɪˈvɑlv/, randomly /ˈɹændm̩li/, alter /ˈɔl.təɹ/

如何用这段视频练习

先完整听一遍视频，不要开口，记下不认识的单词。
先用 0.75 倍速逐句跟读，熟练之后再回到正常速度。
录下自己的声音并与原声对比，特别注意 vector, scalar, dimensional 这类单词。

什么是跟读法？

跟读法 (Shadowing) 是一种有科学依据的语言学习技巧，最初开发用于专业口译员的培训，并由多语言者Alexander Arguelles博士普及。这个方法简单而强大：您在听英语母语原声的同时立即大声重复——就像是一个延迟1-2秒紧跟说话者的影子。与被动听力或语法练习不同，跟读法强迫您的大脑和口腔肌肉同时处理并模仿真实的讲话模式。研究表明它能显着提高发音准确性，语调，节奏，连读，听力理解和口语流利度——使其成为雅思口语备考和真实英语交流最有效的方法之一。

影子跟读法: 阅读完整分步指南 →

跟读练习: Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra - 通过视频学习英语口语

选择练习模式

下载应用

同频道

The inscribed square problem

The Newton art puzzle

How might LLMs store facts | Deep Learning Chapter 7

Transformers, the tech behind LLMs | Deep Learning Chapter 5

Fourier series

Euler's Formula Poem

Vectors | Chapter 1, Essence of linear algebra

Essence of linear algebra preview

Large Language Models explained briefly

Oh, wait, actually the best Wordle opener is not “crane”…

Hilbert's Curve: Is infinite math useful?

But what is a neural network? | Deep learning chapter 1

Divergence and curl: The language of Maxwell's equations, fluid flow, and more

But what is quantum computing? (Grover's Algorithm)

Tie random ends: How many loops?

推荐视频

Derivative as slope of a tangent line | Taking derivatives | Differential Calculus | Khan Academy

Convergence and Divergence: The Return of Sequences and Series

Lec 1 | MIT 18.06 Linear Algebra, Spring 2005

СПИДРАН НА ВСЮ ФИЛОСОФИЮ ЗА 45 МИНУТ

Newton, Leibniz, and Usain Bolt | Derivatives introduction | AP Calculus AB | Khan Academy

Writing Systems: Crash Course Linguistics #16

Lesson Eight: Placement of the Orchestra, Leonard Slatkin's Conducting School

Eco 155: Principles of Macroeconomics Class 1

Orbital Mechanics - Newton's Laws

How I Learned Procedural Generation