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。

there is a separate line in a different direction,

如果一個3-維的變換把空間壓縮到一個平面，

full of vectors that get squished onto the origin.

也滿是經過原點的矢量。

If a 3D transformation squishes space onto a plane,

如果一個3-維的變換把所有的空間壓縮到一根綫上，

there's also a full line of vectors that land on the origin.

然後就滿是經過原點的矢量。

If a 3D transformation squishes all the space onto a line,

這個經過遠點的矢量集叫做矩陣的“0空間(null space)”或者“核(kernel)”

then there's a whole plane full of vectors that land on the origin.

在它們經過0的矢量的意義上，

This set of vectors that lands on the origin is called the "null space" or the "kernel"

它是所有矢量成爲0的空間。

of your matrix.

在綫性方程系統中，就是v 恰好是0矢量。

It's the space of all vectors that become null,

0空間給出方程所有可能的解。

in the sense that they land on the zero vector.

那就是一個很高度的概述

In terms of the linear system of equations, when v happens to be the zero vector,

怎樣來把綫性方程組用幾何方法來考慮。

the null space gives you all of the possible solutions to the equation.

各個系統都和某種類型的綫性變換相聯係的

So that's a very high-level overview

而如果那個變換有一個逆矩陣，

of how to think about linear systems of equations geometrically.

你就可以用那個匿矩陣來解你的系統（綫性方程組）。

Each system has some kind of linear transformation associated

否則，列空間的想法讓我們知道是否存在著一個解，

with it,

而一個0空間的想法幫助我們來理解

and when that transformation has an inverse,

所有可能有的解的集合看起來可以什麽樣的。

you can use that inverse to solve your system.

再提一下我還有很多在這裏還沒有講到了，

Otherwise, the idea of column space lets us understand when a solution even exists,

最明顯的是怎樣來計算這些東西。

and the idea of a null space helps us to understand what the set of

我還不得不來限制用在例子中

all possible solutions can look like.

方程的數目等於未知數的數目。

Again there's a lot that I haven't covered here,

然後這裏的目的並不是試圖來教所有的東西；

most notably how to compute these things.

而是你可以對逆矩陣，列空間，

I also had to limit my scope to examples where the number of equations

和0空間獲得很強的直覺感

equals the number of unknowns.

以及那些直覺使你在將來的學習中更獲益。

But the goal here is not to try to teach everything;

下一個錄像，按大衆的要求，將是對非方矩陣的一個簡單的注解。

it's that you come away with a strong intuition for inverse matrices, column

之後，我將講點我對付點乘積(dot product)，

space, and null space,

以及你們會看到的

and that those intuitions make any future learning that you do more fruitful.

在綫性代數裏一些好東西。

Next video, by popular request, will be a brief footnote about nonsquare matrices.

回頭見！

Then, after that, I'm going to give you my take on dot products,

and something pretty cool that happens when you view them

under the light of linear transformations.

See you then!