## 字幕列表 影片播放

• Hello, hello again.

哈囖，哈囖。

• 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？

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