I described vector coordinates,

乘以係數的法想法一起，我描述了矢量的坐標，

where's this back and forth between, for example, pairs of numbers and two-dimensional vectors.

在那裏兩者之間的來來囘囘，例如，在一對數字和一個2-維矢量之間。

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

現在，我想象中矢量坐標對你們中許多人都已熟悉

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

但是有另一種方法來考慮這些坐標，

which is pretty central to linear algebra.

它對綫性代數的是很核心。

When you have a pair of numbers that's meant to describe a vector, like [3, -2],

如果你有一對數字它們是想用來描述一個矢量的，像[3,-2]，

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

而我想要你們把每個坐標考慮成一個標量，

meaning, think about how each one stretches or squishes vectors,

意思是，想一想每一個怎樣伸長或者壓縮矢量。

In the xy-coordinate system, there are two very special vectors:

在xy-坐標系統中，有兩個非常特殊的矢量：

the one pointing to the right with length 1, commonly called "i-hat", or the unit vector

一個長度為1指向右，通常叫做“i-hat”，

in the x-direction,

或者在x-方向上的單位矢量，

and the one pointing straight up, with length 1, commonly called "j-hat",

而另一個長度為1指向上面的，通常叫做“j-hat” ，或者在y方向上的單位矢量。

or the unit vector in the y-direction.

現在，把我們矢量的x-坐標想作i-hat定量刻度的一個標量，按3倍的因素拉長

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

而y--軸上的坐標作爲 為j-hat作定量刻度的一個標量並拉長2倍並反一個方向

it by a factor of 3,

這這個意義上，這些坐標所描述是兩個經係數修改過的矢量之和。

and the y-coordinate as a scalar that scales j-hat, flipping it and stretching it by a

那可是一個令人驚訝的重要概念，把兩個乘過係數的矢量相加的想法。

factor of 2.

順便提一下，這些兩個矢量，i-hat，和j-hat，有一個特別的名字

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

兩個在一起，它們被稱作一個坐標的基本單位（basis）。

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

這是什麽意思，基本上來說，就是在你把坐標考慮成一些標量的時候，

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

單元矢量就是那些刻度標量，實際上，你們知道，係數。

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

還有一個更技術性的定義，但我將在以後來談。

What this means, basically, is that when you think about coordinates as scalars,

以這兩個特殊的單元矢量來構建我們的坐標系統，

the basis vectors are what those scalars actually, you know, scale.

它提出了一個相當有趣，和深刻的觀點：

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

我們可以選擇不同的單元矢量並得到一個完全合理的新的坐標系統的。

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

00:02:02,806 --> 00:02:07,986 例如，就取一個朝上朝右的矢量一起和一個朝下朝右的。

it raises a pretty interesting, and subtle, point:

化點時間來想一下通過選擇兩個不同的標量係數，用每一個來刻度一個矢量，

We could've chosen different basis vectors, and gotten a completely reasonable, new coordinate

然後加起來，你可以得到所有的矢量。

system.

什麽樣的2-維矢量你可以用變化係數的選擇來達到呢？

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

這答案是你可以達到每一個可能有的2-維矢量，

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

而我想這是一個很好的疑問來想一下為什麽？

Take a moment to think about all the different vectors that you can get by choosing two scalars,

像這樣一對新的單位矢量仍給我們一種

using each one to scale one of the vectors, then adding together what you get.

成立的方法

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

在數字對和2-維矢量之間來來囘囘的，

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

但是這種關係肯定和你用更標準的i-hat

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

和j-hat單元是不同的。這我將來要更深入來講的一件事

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

在不同的坐標系統裏的關係問題

between

任何時侯我們用數字來描述矢量，

pairs of numbers and two-dimensional vectors,

這取決矢量暗指用的什麽樣的單元，

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

這叫做這兩個矢量的一個綫性組合。

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

“綫性”這個字是從哪裏來的？

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

為什麽這和綫條有關係呢？

between

嗯，這不是詞源學，而我喜歡來考慮的

different coordinate systems, but for right now, I just want you to appreciate the fact

一個方法是保持提矢量不變而只變另一個矢量的係數，其結果得出來的矢量的

that

箭頭畫出一條直綫。現在，假如你讓兩個標量係數都有自由的變化範圍并且

any time we describe vectors numerically, it depends on an implicit choice of what basis

考慮你所有可能得出的每個矢量。

vectors we're using.

可以發生兩種情況：

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

對大多數的一對矢量，你將可以達到在平面中每一個可能的點上。

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

每一個2-維矢量都在於你控制之中。

Where does this word "linear" come from?

但是在運氣不好的情況下原先的兩個矢量箭頭的方向都只限于在一根綫上，這就成了一根穿過原點的綫條。

Why does this have anything to do with lines?

實際從技術上來說還有第三種的可能性，你有兩個矢量都可能是0，

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

這樣你就不得不陷在原點上了。

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

還有更多的術語：

the tip of the resulting vector draws a straight line.

對你可以達到的所有可能的矢量加上對給出的矢量的一個綫性組合可以得到的矢量的集合

Now, if you let both scalars range freely, and consider every possible vector that you

叫做這些兩個矢量的伸展（span）.

can get,

因此，重說一下我們剛看到的這一說法，

there are two things that can happen:

大多數2-維矢量的數字對的伸展就是在2-維空間裏所有的矢量。

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

但是在它們排起來的時候，它們的span（伸展）就是所有的矢量，其箭頭都在某一根綫上。

every two-dimensional vector is within your grasp.

還記得我說過綫性代數環繞著矢量相加和比例係數的乘法。

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

好吧，兩個矢量的伸展基本上就是一方法來問

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

"用這兩個基本運算，矢量相加和與係數的相乘

origin.

你可以得到所有的什麽樣可能有的矢量？“

Actually, technically there's a third possibility too:

這是個好時機來談到人們通常把一些矢量看成一些點。

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

把全部在一根綫上的矢量的集合想一下

Here's some more terminology:

那這真是太擠了，而在同時再想一下

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

所有佔滿這平面的的那些2-維矢量，那就更擠了。

pair of vectors

所以我們在處理像這樣的一些矢量的集合時侯，

is called the span of those two vectors.

這是很經常來代表這些矢量，和平常一樣，我要你把那個矢量的箭頭看成一個點而它的箭尾在原點上。

So, restating what we just saw in this lingo,

這樣如果你來想可能有的所有箭頭都在

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

某一根綫上矢量就想這根綫的本身就行了。

but when they line up, their span is all vectors whose tip sits on a certain line.

與此類似，在同一個時閒來想所有可能有的2-維的矢量

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

在概念上就把每一個矢量它的箭頭作爲一個點。

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

這樣，在效果上，對這你將來想到的

"What are all the possible vectors you can reach using only these two fundamental operations,

是無限，2-維的一張平紙的本身，

vector addition and scalar multiplication?"

而把箭頭從紙上拿掉了。

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

一般來說，如果你在處理矢量的集合

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

時，把它們對想成一些點，這就比較方便。

and more crowded still to think about all two-dimensional vectors all at once, filling

因此對我們span（伸展）的例子，大多數的矢量伸展的結果就是整張無限

up the plane.

的紙，整個2-維空間，但如果它們在同一條綫上，它們的伸展結果就只是一條綫了

So when dealing with collections of vectors like this,

伸展的思路變得更為有趣了如果我們開始考慮在一個三維空間裏的矢量。

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

例如，如果你在3維空間裏取二個不在同一方向上的矢量，

The point at the tip of that vector, where, as usual, I want you thinking about that vector

把它們的伸展是什麽意思呢？

with its tail on the origin.

嗯，它們的伸展是這兩個矢量所有可能有的綫性組合，意思是所有你所得到的

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

以某些方法通過對每個矢量乘以係數，然後把它們向加起來。

line,

你可能想象一下轉動兩個旋鈕來改變

just think about the line itself.

定義其綫性組合，把這乘過係數的矢量並跟隨著合成後矢量的箭頭。

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

那個箭頭將畫出某種平面的紙，穿過

conceptualize each one as the point where its tip sits.

三維空間的原點。

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

或者跟精確地來說，所以可能有的矢量那些箭頭在那張平面的紙上

space itself,

就是你們那兩個矢量的伸展。

leaving the arrows out of it.

這樣，如果我們加上第三個矢量然後我們來考慮這三個傢夥會怎麽樣呢？

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

三個矢量的綫性組合的定義基本和兩個的是同樣的方法；

and if you're dealing with a collection of vectors, it's convenient to think of them

你選擇三個不同的係數，然後再加起來。

all as points.

而又是一次，這些矢量的伸展是所有可能有的綫性組合的集合。

So, for our span example, the span of most pairs of vectors ends up being

在這裏可能發生兩個不同的事：

the entire infinite sheet of two-dimensional space,

如果你的那個第三個矢量正好是在前兩個的伸展上

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

那麽這個伸展並沒有發生改變；你就像是被關在那個平的紙上了一樣。

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

換句話說，把一個乘過係數的第三個矢量加到這綫性組合

space.

並沒有真正地讓你接近任何新的矢量。

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

但是如果你只是隨便選一個第三個矢量，這幾乎肯定不在

what does it mean to take their span?

那前兩個的伸展上的。

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

然後既然它是指著一個不同的方向

vectors, meaning

它為接近每一個可能的3-維矢量解了鎖。

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

它圍著第一和第二個（矢量）的伸展平面移動，將它掃過所有的空間。

them together.

另一種方法來考慮是你充分利用供你使用的係數，自由地改變著來

You can kind of imagine turning two different knobs to change the two scalars defining the

接近完全的3-維空間。

linear combination,

現在，在這樣的情況下，第三個矢量已經在前兩個的伸展上了

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

或者那兩個矢量正好在同一條綫上

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

我們想要什麽術語來描述這個事實

space.

至少這些矢量中的一個是重復的-沒有為我們的伸展加上什麽。

This flat sheet is the span of the two vectors,

每次發生這樣的，你有多個矢量而你

or more precisely, the set of all possible vectors whose tips sit on that flat sheet

可以拿掉一個而沒有影響到這伸展，

is the span of your two vectors.

這有關的術語是“綫性上依賴的”，

Isn't that a beautiful mental image?

一種其它（矢量）的綫性組合因爲它已經在其它（矢量）的伸展之中了。

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

在另一個方面，如果每個矢量確實是對這伸展另外加上了一維的話

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

它們就被稱爲“綫性上獨立的”。

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

這樣有了所有的那種術語，并且也希望與它一起在腦子裏對它有些形象。

together.

在我們分手前讓我留給你些猜想。

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

一個空間的基本矢量是一組伸展

Two different things could happen here:

那個空間的綫性上獨立的矢量。

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

現在，給出了我剛剛那樣的描述

then the span doesn't change; you're sort of trapped on that same flat sheet.

想一下為什麽這個定義會有道理的。

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 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!