So, moving forward

這樣，接著講下去

I will be assuming you have a visual understanding of linear transformations

我將假定你對綫性變換有一個視覺上的理解

and how they're represented with matrices

以及他們是怎樣用矩陣來表示的

the way I have been talking about in the last few videos.

這方法我在過去的幾個錄像中講過的。

If you think about a couple of these linear transformations

如果你想一下這些綫性變換

you might notice how some of them seem to stretch space out

你也許會注意到有些是怎樣看起來在空間裏拉伸著

while others squish it on in.

而另外一些把它壓縮進去。

One thing that turns out to be pretty useful to understanding one of these transformations

一個東西卻對理解這些變換中的一個相當有用的

is to measure exactly how much it stretches or squishes things.

是來精確地度量它把那些東西拉伸或者壓縮了多少。

More specifically

更具體來度量對給出的區域

to measure the factor by which the given region increases or decreases.

增加或者減少的係數。

For example

舉個例子

look at the matrix with the columns 3, 0 and 0, 2

看一個矩陣它的列是(3,0) 和(0,2)

It scales i-hat by a factor of 3

它對i-hat乘以一個係數3

and scales j-hat by a factor of 2

並對j-hat乘以一個係數

Now, if we focus our attention on the one by one square

現在，如果我們把注意力集中在1X1 的方塊s

whose bottom sits on i-hat and whose left side sits on j-hat.

它的底部在i-hat是而左面在j-hat上。

After the transformation, this turns into a 2 by 3 rectangle.

轉換之後，這就變成一個2X3 的長方塊。

Since this region started out with area 1, and ended up with area 6

因爲這個區域開始的面積是1，而面積變成了6

we can say the linear transformation has scaled it's area by a factor of 6.

我們可以是這綫性變換已經通過一個係數6來放大了它的面積

Compare that to a shear

將它與一個剪切來比較

whose matrix has columns 1, 0 and 1, 1.

剪切的矩陣的列是(1,0)和(1,1).

Meaning, i-hat stays in place and j-hat moves over to 1, 1.

意思是，h-hat停在原地而j-hat移動到(1,1)。

That same unit square determined by i-hat and j-hat

那個同樣的由i-hat和j-hat所決定的單位方塊

gets slanted and turned into a parallelogram.

被歪掉了並成了一個平行四邊形。

But, the area of that parallelogram is still 1

但是，那個平行四邊形的面積仍是1

since it's base and height each continue to each have length 1.

因爲它的底綫和高度繼續有長度1。

So, even though this transformation smushes things about

因此即使這個變換壓變了這東西

it seems to leave areas unchanged.

它看來面積到沒變。

At least, in the case of that one unit square.

至少，在一個單位方塊的情況下。

Actually though

雖然實際上

if you know how much the area of that one single unit square changes

如果你知道一個單位方塊的面積變化了多少的話

it can tell you how any possible region in space changes.

這也能告訴你在空間裏的任何區域的變化的。

For starters

這麽開頭吧

notice that whatever happens to one square in the grid

注意在網格裏的一個方塊不管已經在

has to happen in any other square in the grid

任何的在網格裏發生了怎麽樣的變化

no matter the size.

尺寸大小是無關緊要的。

This follows from the fact that grid lines remain parallel and evenly spaced.

這個出自這樣的事實網格綫一直平行並間隔均等的。

Then, any shape that is not a grid square

然後，任何不是一個方格形狀的

can be approximated by grid squares really well.

可以用網格來很好地近似的。

With arbitrarily good approximations if you use small enough grid squares.

如果你用足夠小的方塊就有隨便樣什麽近似程度。

So, since the areas of all those tiny grid squares are being scaled by some single amount

這樣，因爲所有的那些很小的方塊都以同樣的一個係數被縮小的區域

the area of the blob as a whole

這一團作爲一個整體也將

will also be scaled also by that same single amount.

被縮小一個同樣的係數。

This very special scaling factor

這個非常特殊的縮小的係數

the factor by which a linear transformation changes any area

一個綫性變換以那個係數改變了

is called the determinant of that transformation.

任何面積的叫做那個變換的行列式值。

I'll show how to compute the determinate of a transformation using it's matrix later on

在這個錄像後面我將演示給你看

in the video

怎樣用來計算一個變換的行列式值，

but understanding what it is, trust me, much more important than understanding the computation.

但是懂得它是什麽，信任我，比懂得計算更我重要。

For example the determinant of a transformation would be 3

例如一個變換的行列式值會是3

if that transformation increases the area of the region by a factor of 3.

如果那個變換增加了那個區域的面積3倍。

The determinant of a transformation would be 1/2

這行列式值會是1/2

if it squishes down all areas by a factor of 1/2.

如果它把面積壓縮到1/2.

And, the determinant of a 2-D transformation is 0

而，一個2-維變換的行列式是0

if it squishes all of space onto a line.

如果把所以的空間壓到一根綫上。

Or, even onto a single point.

或者，甚至縮到一個點上。

Since then, the area of any region would become 0.

然後，任何區域的面積就會成爲0了。

That last example proved to be pretty important

最後那個例子證明是非常重要的。

it means checking if the determinant of a given matrix is 0

它意味著檢查如果一個給出的矩陣的行列式值是不是0

will give away if computing weather or not the transformation associated with that matrix

將給出和那個矩陣有關的變換是不是

squishes everything into a smaller dimension.

把所有的東西壓縮進了一個更小的尺寸。

You will see in the next few videos

你們在以後的幾個錄像中將知道

why this is even a useful thing to think about.

爲什麽這甚至還是一個有用的東西來考慮一下的。

But for now, I just want to lay down all of the visual intuition

但是現在，我只想寫下所有的視覺上的直覺

which, in and of itself, is a beautiful thing to think about.

這個本身，就是一件美麗的東西來想一想的。

Ok, I need to confess that what I've said so far is not quite right.

Ok，我需要坦白我剛已說過的並不很正確。

The full concept of the determinant allows for negative values.

行列式值的完整概念允許有負數。

But, what would scaling an area by a negative amount even mean?

但是，把一個面積放大縮小一個負數還會有什麽意思？

This has to do with the idea of orientation.

這個不得不和方向的概念有關係。

For example

例如

notice how this transformation

注意這個變換

gives the sensation of flipping space over.

給出把空間翻個身的感覺。

If you were thinking of 2-D space as a sheet of paper

如果你把2-維空間當作一張紙，

a transformation like that one seems to turn over that sheet onto the other side.

像那樣的一個變換一個人看起來像是把紙翻到另一個面了。

Any transformations that do this are said to "invert the orientation of space."

任何作這樣的變換的就說成是來“反轉空間的定向。”

Another way to think about it is in terms of i-hat and j-hat.

另一個方法來考慮是用i-hat和j-hat。

Notice that in their starting positions, j-hat is to the left of i-hat.

注意到在它們開始的位置，j-hat是在 i-hat的左面的。

If, after a transformation, j-hat is now on the right of i-hat

如果在轉換之後，j-hat到了i-hat的右面，

the orientation of space has been inverted.

空間的定向已被反了過來。

Whenever this happens

任何時候發生了這個

whenever the orientation of space is inverted

任何時候空間的定向被翻轉了

the determinant will be negative.

這行列式值將是個負數。

The absolute value of the determinant though

這行列式值的絕對值

still tells you the factor by which areas have been scaled.

仍告訴你這個面積被放大縮小的係數。

For example

例如

the matrix with columns 1, 1 and 2, -1

一個矩陣的列分別是(1,1)和(2,-1)

encodes a transformation that has determinant

編碼著一個變換，它的行列式值

Ill just tell you

我告訴你

-3.

是-3。

And what this means is

而它的意思是

that, space gets flipped over

空間被翻了一個身

and areas are scaled by a factor of 3.

并且面積放大了3倍。

So why would this idea of a negative area scaling factor

那麽為什麽一個負的面積放大縮小係數

be a natural way to describe orientation flipping?

會是一種自然的方法來描述定向的翻轉？

Think about the seres of transformations you get

考慮一下你有這一系列的變換

by slowly letting i-hat get closer and closer to j-hat.

慢慢的讓i-hat 越來越結局 j-hat。

As i-hat gets closer

隨著i-hat 的接近

all the areas in space are getting squished more and more

空間裏所有的面積越來越被壓縮

meaning the determinant approaches 0.

意思是行列式值接近於0.

once i-hat lines up perfectly with j-hat,

一旦 i-hat 和 j-hap完全重合，

the determinant is 0.

行列式值就是0.

Then, if i-hat continues the way it was going

然後，如果i-hat 繼續這樣走下去

doesn't it kinda feel natural for the determinant to keep decreasing into the negative numbers?

這不正是讓行列式值減小進入負數那樣感到很自然的嗎？

So, that is the understanding of determinants in 2 dimensions

因此，那就是在2-維空間對行列式值的理解

what do you think it should mean for 3 dimensions?

對3-維你想它應該怎樣來想呢？

It [determinant of 3x3 matrix] also tells you how much a transformation scales things

如果3x3 矩陣的行列式也告訴你一個變換對一些東西放大縮小了多少

but this time

但這次

it tells you how much volumes get scaled.

它告訴你多少體積被放大縮小了

Just as in 2 dimensions

就像在2-維的那個一樣

where this is easiest to think about by focusing on one particular square with an area 1

集中是一個特殊的一個面積為1的方塊

and watching only what happens to it

並只看著對在3-維空間裏它所發生的，

in 3 dimensions

這會容易一些

it helps to focus your attention

它有助於把你的注意力集中

on the specific 1 by 1 by 1 cube

在這個特定的1x1x1方塊

whose edges are resting on the basis vectors

它的邊在基本矢量

i-hat, j-hat, and k-hat.

i-hat, j-hat, 和 k-hat 上.

After the transformation

轉換之後

that cube might get warped into some kind of slanty slanty cube

那個方塊可能被扭曲到像是很斜很斜的方塊

this shape by the way has the best name ever

順便提一下，這個形狀有一個最好的

parallelepiped.

名字parallelepiped（平行管柱）

A name made even more delightful when your professor has a nice thick Russian accent.

一個名字甚至更開心的如果你的教授有一個很強的俄羅斯口音。

Since this cube starts out with a volume of 1

既然這個方塊的體積為1

and the determinant gives the factor by which any volume is scaled

而行列式值給出體積被放大的係數

you can think of the determinant

你可以把行列式值簡單的考慮成

as simply being the volume of that parallelepiped

那方塊變成的

that the cube turns into.

那個parallepiped的體積。

A determinate of 0

一個是0的行列式值

would mean that, all of space is squished onto something with 0 volume

將意味著，所有的空間都被壓縮的一個

meaning ether a flat plane, a line, or in the most extreme case

0體積的什麽東西，意思是或者是一個平面，一條綫，或者在極端情況下

onto a single point.

成爲一個點。

Those of you who watched chapter 2

你們那些看過第二章的

will recognize this as meaning

將認識到這個意思

that the columns of the matrix are linearly dependent.

矩陣的列是綫性相關的。

Can you see why?

你能知道為什麽嗎？

What about negative determinants?

那麽負的行列式值呢？

What should that mean for 3 dimensions?

那麽對3-維的應該是什麽意思呢？

One way to describe orientation in 3-D

在3-維空間來描述方向的一個方法是

is with the right hand rule.

用右手定則。

Point the forefinger of your right hand

用你的右手的食指指向

in the direction of i-hat

i-hat 的方向

stick out your middle finger in the direction of j-hat

在j-hat 方向上伸出你的中指

and notice how when you point your thumb up

而注意到你的姆指朝上

it is in the direction of k-hat.

它就是k-hat 的方向。

If you can still do that after the transformation

在變換之後如果你仍能夠做這個的，

orientation has not changed

定向沒有變化

and the determinant is positive.

那麽行列式值是正的；

Otherwise

否則

if after the transformation it only makes since to do that with your left hand

如果在變換之後，你只能用你的左手來做的話

orientation has been flipped

定向被翻過來了

and the determinant is negative.

而行列式值是負的。

So if you haven't seen it before

如果你以前不知道

you are probably wondering by now

你也許在想

"How do you actually compute the determinant?"

“那你實際上是怎樣來計算行列式值的呢？”

For a 2 by 2 matrix with entries a, b, c, d

對一個2x2的矩陣它的項為a, b, c, d

the formula is (a * d) - (b * c).

這公式是(a*d) - (b*c)。

Here's part of an intuition for where this formula comes from

這個公式是怎麽來的這裏是一部分的直覺

lets say the terms b and c both happed to be 0.

讓我們假定項數 b 和 c 都正好是0.

Then the term a tells you how much i-hat is stretched in the x direction

然後項數 a 告訴你 i-hat 在x方向

and the term d

上伸出多少而項數 d 告訴你

tells you how much j-hat is stretched in the y direction.

j-hat 在y 方向是伸出多少。

So, since those other terms are 0

這樣，因爲其它的項都是0

it should make sense that a * d

這應該可以理解 a*d 給出我們最喜歡的

gives the area of the rectangle that our favorite unit square turns into.

單位方形所變成的矩形的面積。

Kinda like the 3, 0, 0, 2 example from earlier.

有點像是早些的例子 3, 0, 0, 2

even if only one of b or c are 0

即使 b, 或者 c 當中只有一個是0

you'll have a parallelogram

你就有一個平行四邊形

with a base a

它的底是 a

and a height d.

而高為 d。

So, the area should still be

這樣，這面積仍應該是

a times d.

a 乘以 d。

Loosely speaking

不太嚴格地說

if both b and c are non-0

如果 b 和 c 都不是0

then that b * c term

然後 b*c 那個項告訴你

tells you how much this parallelogram

這個平行四邊形被拉伸看多少

is stretched or squished in the diagonal direction.

或者在對角綫方向說被壓縮了多少。

For those of you hungry for a more precipice description of this b * c term

對你們當中有些急於要對 b*c 這個項有一個更精確的描述

here's a helpful diagram if you would like to pause and ponder.

這裏有一幅有幫助的圖如果你想要來停等一下並想一下的話。

Now if you feel like computing determinants by hand

現在如果你感到 喜歡用手算行列式值的話

is something that you need to know

是一件什麽事情你需要了知道的

the only way to get it down is to just

唯一來做的方法

practice it with a few.

就用它來做幾個練習。

There's not really that much I can say or animate that is going to drill in the computation.

真的我也沒有很多可說的或者動畫一下要來計算的。

This is all tripply true for 3-rd dimensional determinants.

對3-維的行列式值都是三個的

There is a formula [for that]

對此有一個公式

and if you feel like that is something you need to know

而如果你覺得想要啦知道的話

you should practice with a few matrices

你應該做幾個矩陣的練習

or you know, go watch Sal Kahn work through a few.

或者你知道的，去看看Sal Kahn 做的幾個。

Honestly though

雖然老實說

I don't think those computations fall within the essence of linear algebra

我不認爲那些計算在綫性代數的範圍裏的

but I definitely think that knowing what the determinate represents

但是我肯定認爲行列式值所代表的

falls within that essence.

是在那個精要中的。

Here's kind of a fun question to think about before the next video

在下一個錄像之前有一個像是有趣的問題來想一下

if you multiply 2 matrices together

如果你把兩個矩陣相乘起來

the determinant of the resulting matrix

其結果得到的矩陣的行列式值

is the same as the product of the determinants of the original two matrices

是和兩個原來的矩陣的行列式值的乘積是相同的。

if you tried to justify this with numbers

如果你用數字來下證明

it would take a really long time

它會化好多時間的

but see if you can explain why this makes sense in just one sentence.

但是看看如果你可以解釋這點只用一句話就可以懂的話。

Next up

下面

I'll be relating the idea of linear transformations covered so far

我將把至今所講過的綫性變換的思想

to one of the areas where linear algebra is most useful

來聯係到綫性代數最有用的

linear systems of equations

綫性方程組上。

see ya then!

到時再見！