## 字幕列表 影片播放

• As you can probably tell by now, the bulk of this series is on understanding matrix

正如你們現在也許可以講得出，這個系列，

• and vector operations

通過更多綫性變換的視覺透鏡，

• through that more visual lens of linear transformations.

而在於懂得j矩陣和矢量的各種運算。

• This video is no exception, describing the concepts of inverse matrices,

這個錄像也不例外，通過那個透鏡描述

• column space, rank, and null space through that lens.

反矢量，列空間，秩，和零空間的概念。

• A forewarning though: I'm not going to talk about the methods for actually computing these

但有個預先警告：我不講實際計算它們的

• things,

而有些人會說那開始相當重要的。

• and some would argue that that's pretty important.

在這個系列之外有很多來學習那些方法很好的資源。

• There are a lot of very good resources for learning those methods outside this series.

查關鍵詞：“Gaussian elimination(高司消除法)”和“Row echolon form”

• Keywords: "Gaussian elimination" and "Row echelon form."

我想在我實際加在這裏的大部分的價值是在直覺方面的。

• I think most of the value that I actually have to add here is on the intuition half.

再加上，在實際中，我們可是通常用軟件來計算這些東西的。

• Plus, in practice, we usually get software to compute this stuff for us anyway.

首先，講幾句綫性代數的用処。

• First, a few words on the usefulness of linear algebra.

到了現在，你們已經對它怎樣

• By now, you already have a hint for how it's used in describing the the manipulation of

用在描述空間上有了一點提示

• space,

那就是對像計算機圖象和機器人之類的東西是有用的。

• which is useful for things like computer graphics and robotics,

但綫性代數是更廣汎的用處和幾乎

• but one of the main reasons that linear algebra is more broadly applicable,

任何的技術學科都需要的，

• and required for just about any technical discipline,

是它讓我們來解某些方程組。

• is that it lets us solve certain systems of equations.

在我說到“方程組”的時候，我的

• When I say "system of equations," I mean you have a list of variables, things you don't

意思是你有一批變量，你所不知道的

• know,

東西，和一組把它們聯係起來的方程。

• and a list of equations relating them.

在很多情況下，這些方程可以是很複雜的，

• In a lot of situations, those equations can get very complicated,

但是，如果你運氣好，他們可以是某種特殊的形式。

• but, if you're lucky, they might take on a certain special form.

在每個方程裏，對每個變量唯一做的事

• Within each equation, the only thing happening to each variable is that it's scaled by some

就是乘以了一個不變的係數，

• constant,

而對每一個乘了係數的變量唯一做的事

• and the only thing happening to each of those scaled variables is that they're added to

它們互相相加。

• each other.

因此，沒有指數或者花妙的函數，或者把兩個變量相乘起來那樣的事情。

• So, no exponents or fancy functions, or multiplying two variables together; things like that.

組織這樣的特殊的方程系統的典型方法

• The typical way to organize this sort of special system of equations

是把所有變量放到左邊，

• is to throw all the variables on the left,

並把餘下的常數放在右面。

• and put any lingering constants on the right.

把同樣的變量豎直對好也是很好的，

• It's also nice to vertically line up the common variables,

而這樣做的你也許需要把一些係數

• and to do that you might need to throw in some zero coefficients whenever the variable

為0的放到在方程中沒有的變量。

• doesn't show up in one of the equations.

這叫做一個“綫性方程系統。”

• This is called a "linear system of equations."

你們也許會注意到這個看上去很像

• You might notice that this looks a lot like matrix vector multiplication.

矩陣矢量乘法。事實上，你可以把所有的方程放進一個矢量方程，在其中你

• In fact, you can package all of the equations together into a single vector equation,

有這矩陣包括了所有的不變的係數，和

• where you have the matrix containing all of the constant coefficients, and a vector containing

一個矢量包括了所有的變量，

• all of the variables,

而它們的矩陣和矢量的積等於一些常數的矢量。

• and their matrix vector product equals some different constant vector.

讓我們命名常數矩陣為A，

• Let's name that constant matrix A,

代表變量的矢量為x，

• denote the vector holding the variables with a boldface x,

並把在右邊的常數矢量叫做v。

• and call the constant vector on the right-hand side v.

這個不只是為了要把我們的方程系統

• This is more than just a notational trick to get our system of equations written on

寫成一個等式的一個字面上的一個技巧而已。

• one line.

它對這問題的一種很美妙幾何解釋更容易懂得。

• It sheds light on a pretty cool geometric interpretation for the problem.

矩陣A對應以一些綫性變換，因此解

• The matrix A corresponds with some linear transformation, so solving Ax = v

Ax = v 意思是我們在找一個矢量 x

• means we're looking for a vector x which, after applying the transformation, lands on

對它施加這變換之後，就停在v。

• v.

想一想這裏發生著什麽。

• Think about what's happening here for a moment.

你可以有這個真正複雜的所有互相混在一起的多個變量

• You can hold in your head this really complicated idea of multiple variables all intermingling

的想法在你的頭腦裏

• with each other

就想著空間的移動和變化來得出一個

• just by thinking about squishing and morphing space and trying to figure out which vector

矢量停留住的地方。

• lands on another.

真好，對嗎？

• Cool, right?

開始簡單的，讓我們有一個有兩個等式和兩個未知數的。

• To start simple, let's say you have a system with two equations and two unknowns.

那意思是矩陣 A 是一個2x2的矩陣，

• This means that the matrix A is a 2x2 matrix,

而 v 和 x 都是一個2維的矢量。

• and v and x are each two dimensional vectors.

現在，我們怎樣來考慮這個方程的解

• Now, how we think about the solutions to this equation

取決於和A相關的變換是否把所有的

• depends on whether the transformation associated with A squishes all of space into a lower

變到一個更低維數的空間，

• dimension,

像一根綫或者一個點，或者它擴展

• like a line or a point,

它所在的2維空間。

• or if it leaves everything spanning the full two dimensions where it started.

用說一個錄像中的的說法，我們再把

• In the language of the last video, we subdivide into the case where A has zero determinant,

這情況細分而在A ，和A的行列式值不為0 的情況。

• and the case where A has nonzero determinant.

讓我們以最為可能的情況說起，那就是行列式值不為0，

• Let's start with the most likely case, where the determinant is nonzero,

意思是空間沒有被壓成一個沒有面積的區域。

• meaning space does not get squished into a zero area region.

這這種情況下，總歸有一個并且只有一個矢量 在 v上，

• In this case, there will always be one and only one vector that lands on v,

而你可以用逆變換來找到它的。

• and you can find it by playing the transformation in reverse.

而我們像這樣地重新回放磁帶，

• Following where v goes as we rewind the tape like this,

你將發現矢量 x 而A乘以 x 等於v。

• you'll find the vector x such that A times x equals v.

在你施加一個逆變換的時候，它實際上

• When you play the transformation in reverse, it actually corresponds to a separate linear

相當於另一個不同的變換，

• transformation,

通常稱作“A的逆（矩陣）”

• commonly called "the inverse of A"

記法為A的負一次方。

• denoted A to the negative one.

例如，如果A是一個90度的逆時針轉動的話，

• For example, if A was a counterclockwise rotation by 90º

那麽A的逆矩陣將是順時針轉動90度。

• then the inverse of A would be a clockwise rotation by 90º.

如果A是向右的剪切，那就是把j-hat推向右面一個單位，

• If A was a rightward shear that pushes j-hat one unit to the right,

其逆矩陣將是一個向左的剪切，那就是把j-hat向左推一個單位。

• the inverse of a would be a leftward shear that pushes j-hat one unit to the left.

一般來說，A的逆矩陣是矩陣A的一種

• In general, A inverse is the unique transformation with the property that if you first apply

獨特的變換，它具有這樣的性質

• A,

如果你施加A，然後施加A的逆矩陣，

• then follow it with the transformation A inverse,

你回到你所開始的地方。

• you end up back where you started.

在施加一個變換之後又施加另一個，那在代數裏就是矩陣的乘法

• Applying one transformation after another is captured algebraically with matrix multiplication,

所以這種A的逆矩陣乘以A的變換的

• so the core property of this transformation A inverse is that A inverse times A

性質的核心相當於什麽也沒有做

• equals the matrix that corresponds to doing nothing.

而什麽也沒做的變換就叫做“等同變換。”

• The transformation that does nothing is called the "identity transformation."

它把i-hat和j-hat都留在原地，沒有移動過，

• It leaves i-hat and j-hat each where they are, unmoved,

所以它的列是1, 0, 和 0, 1。

• so its columns are one, zero, and zero, one.

一旦你找到這個逆矩陣，在實踐中，你用計算機來做的，

• Once you find this inverse, which, in practice, you do with a computer,

你就可以通過這個逆矩陣乘以矩陣v來解你的方程了。

• you can solve your equation by multiplying this inverse matrix by v.

再說一遍，幾何意義上這意味著你

• And again, what this means geometrically is that you're playing the transformation in

用v 施加著逆轉換。

• reverse, and following v.

這種行列式不是0的情況，這對隨機

• This nonzero determinant case, which for a random choice of matrix is by far the most

選擇的一個矩陣是最有可以的一種，

• likely one,

就像相當於你有2個未知數和2個方程的想法，

• corresponds with the idea that if you have two unknowns and two equations,

在大多數情況下那有一個獨特的解。

• it's almost certainly the case that there's a single, unique solution.

這種想法在更高維數中也是容易理解的，

• This idea also makes sense in higher dimensions,

如果方程的個數等於未知數的個數。

• when the number of equations equals the number of unknowns.

重復一遍，方程系統可以翻譯成用幾何的解釋，

• Again, the system of equations can be translated to the geometric interpretation

你有一些變換， A

• where you have some transformation, A,

和一些矢量 v

• and some vector, v,

并且你在尋找一個矢量 x 它停到 v 上。

• and you're looking for the vector x that lands on v.

只要A的變換不把空間壓縮進一個更低

• As long as the transformation A doesn't squish all of space into a lower dimension,

維數的空間，就是說，它的行列式值

• meaning, its determinant is nonzero,

不為0，將將有一個逆轉換，A的逆矩陣，

• there will be an inverse transformation, A inverse,

具有這樣的性質，如果你先做A

• with the property that if you first do A,

然後你做A的逆轉換，

• then you do A inverse,

它和什麽也不做的效果是一樣的。

• it's the same as doing nothing.

而來解你的方程，你就得把

• And to solve your equation, you just have to multiply that reverse transformation matrix

矢量v 來乘以那個逆矩陣。

• by the vector v.

但是 如果這行列式值為0，那麽

• But when the determinant is zero, and the transformation associated with this system

和這個系統有關的變換

• of equations

把空間壓縮到一個更低維數，就沒有逆變換了。

• squishes space into a smaller dimension, there is no inverse.

你不能去-壓縮一根綫來把它回到一個平面的。

• You cannot un-squish a line to turn it into a plane.

至少那不是一個函數能做的一件事。

• At least, that's not something that a function can do.

那需要轉換各個矢量變成

• That would require transforming each individual vector

都是矢量的一整條綫。

• into a whole line full of vectors.

但是函數只能有一個輸入變成一個輸出。

• But functions can only take a single input to a single output.

與此相似，對於3個未知數的3個方程中，

• Similarly, for three equations in three unknowns,

如果相應的變換把3維的空間壓縮到

• there will be no inverse if the corresponding transformation

這個平面上，或者甚至如果它吧它壓成

• squishes 3D space onto the plane,

一根綫，或者一個點的話，那就沒有逆矩陣了。

• or even if it squishes it onto a line, or a point.

所有的這些都相對應於一個行列式為0的情況，

• Those all correspond to a determinant of zero,

因爲然後的區域被壓縮到一個體積為0的一個東西了。

• since any region is squished into something with zero volume.

如果沒有逆矩陣的話也是有可能存在一個解的，

• It's still possible that a solution exists even when there is no inverse,

這只不過是在你的變換時把空間壓縮成

• it's just that when your transformation squishes space onto, say, a line,

比分說，一根綫，而你運氣必須好到這矢量在那根綫上的一個地方。

• you have to be lucky enough that the vector v lives somewhere on that line.

你可能注意到這些行列式值為0的情況中有些比其他的要更嚴格一些。

• You might notice that some of these zero determinant cases feel a lot more restrictive than others.

舉個例，給出一個3x3 的矩陣，這看起來更難些來存在一個解的

• Given a 3x3 matrix, for example, it seems a lot harder for a solution to exist

如果它和相比一個平面壓縮空間到一條綫相比

• when it squishes space onto a line compared to when it squishes things onto a plane,

即使兩者都是那些行列式為0的。

• even though both of those are zero determinant.

比起我們說“行列式值為0”，我們有些更為確定的語言。

• We have some language that's a bit more specific than just saying "zero determinant."

在一個變換輸出是一根綫的時候，意思是這是1-維的，

• When the output of a transformation is a line, meaning it's one-dimensional,

我們說這變換有‘秩(rank)’為1.

• we say the transformation has a "rank" of one.

如果所有的矢量都在某個2-維平面上，

• If all the vectors land on some two-dimensional plane,

我們是這變換有一個‘秩（rank）’為2.

• We say the transformation has a "rank" of two.

因此'秩(rank)'這個字意思是一個便會的輸出的數字的維數。

• So the word "rank" means the number of dimensions in the output of a transformation.

例如，在一個2x2 矩陣的情況中，秩為2是它所能做到最高的了。

• For instance, in the case of 2x2 matrices, rank 2 is the best that it can be.

它意味著單位矢量繼續擴展到整個2-維空間，

• It means the basis vectors continue to span the full two dimensions of space, and the

而行列式值不為0.

• determinant is nonzero.

但是對3x3矩陣，秩為2意味著我們已經坍縮了，

• But for 3x3 matrices, rank 2 means that we've collapsed,

但並沒有想在一個秩為1的情況會有的那種坍縮。

• but not as much as they would have collapsed for a rank 1 situation.

如果一個3-維的變換具有一個不為0的行列式值，而它的輸出填滿了整個3-維空間，

• If a 3D transformation has a nonzero determinant, and its output fills all of 3D space,

它就有一個秩是3.

• it has a rank of 3.

對你的矩陣所有可能有的輸出的集合，

• This set of all possible outputs for your matrix,

不管它是一條綫，一個平面，3-維空間

• whether it's a line, a plane, 3D space, whatever,

被稱作你的矩陣的“列空間”。

• is called the "column space" of your matrix.

你也許可以猜出這名字是從那兒來的。

• You can probably guess where that name comes from.

你矩陣的列告訴你這單位矢量停在什麽地方的，

• The columns of your matrix tell you where the basis vectors land,

而這些經變換的單位矢量給出所有可能有的輸出。

• and the span of those transformed basis vectors gives you all possible outputs.

換句話說，列空間是你的矩陣的列的擴展。

• In other words, the column space is the span of the columns of your matrix.

因此秩的一個更為精確定義會是

• So, a more precise definition of rank would be that

如果這個秩是它所能達到最高的，

• it's the number of dimensions in the column space.

意思是它等於列的數目，我們叫這矩陣“全秩(full rank)”。

• When this rank is as high as it can be,

注意，0矢量將永遠被包括在這列空間裏。

• meaning it equals the number of columns, we call the matrix "full rank."

因爲綫性變換必須保持原點不動的。

• Notice, the zero vector will always be included in the column space,

對一個全秩變換來說，停在原點的矢量

• since linear transformations must keep the origin fixed in place.

只有是0矢量的自身。

• For a full rank transformation, the only vector that lands at the origin is the zero vector

但是對不是全秩的矩陣，也就是它壓縮到一個更小的維數的，來說

• itself,

你可以有一大堆停在0上的矢量。

• but for matrices that aren't full rank, which squish to a smaller dimension,

如果一個2-維的變換把空間壓縮到一根綫是，舉個例子說，

• you can have a whole bunch of vectors that land on zero.

在一個不同的方向上有著一根分開的綫，

• If a 2D transformation squishes space onto a line, for example,

滿是矢量被壓縮到原點。