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信息不丢矢，左逆有意义：判断逆矩阵存在的简单条件

Matrix inverses make sense: a simple condition for when the inverse exists

In a linear algebra course,

Matrix inverses are something that get a lot of attention.

There are all these formulas thrown at you

like for the determinants, the inverses of a 2×2 matrices

cramner’s rule etc etc.

[好吧 我承认我是查了当年的课本才写得出这些公式 毕竟这些知识早就已经尘封在痛苦记忆的深处了]
These formulas are great,

but I think they can obscure the very simple idea behind

What an inverse actually is, and when it exists.

You’ve probably been taught that the inverse exists

if and only if the determinant isn’t 0.

But most students have no idea

what the determinant has to do with anything

let alone inverses.

[（紫色字体）我看不明白]
I’m going to teach a completely different condition for when an inverse exists,

that I think is much more intuitive.

But before all that:

Do you know what the linear transformation is?

If not, go watch my last video because

none of this one is going to make sense at all otherwise.

In fact, if you want to brush up on vectors and bases,

you can watch the video before that too.

Even though this video is about matrix inverses,

I’m not going to define what an inverse is straight away.

Instead I’m going to define something called a left inverse

because I think understanding these first

will give you a better intuition for inverses themselves.

Imagine you have a bunch of vectors

in some vector space, in this case in 2d,

and you apply some Matrix M on them, here it rotates them.

Then you think you’d like to undo

what you just did and get the vectors back to where they were.

In this example,

what transformation undoes the rotation?

you’d just rotate everything back by the same angle. Right?

That’s all a left inverse is.

It’s the matrix that undoes the original matrix,

so it’s like you’ve done nothing at all.

If I wanted to write this as an equation,

it’d say, if you do M,

then you do L,

It’s the same as if you did nothing.

This thing is called the identity matrix, by the way

and it just means the transformation where you do nothing.

This thing is called the left inverse for… hopefully obvious reasons.

So now we know,

the left inverse is a matrix that undoes the original matrix’s action.

The annoying thing about inverses is really their name.

It sounds like the *inverse* should

be the thing that undoes a matrix.

Instead,the definition of M inverse is:

M inverse undoes M
M的逆可以还原M矩阵
And M undoes M inverse.

Going back to our example:

if you do M first and then L, that’s the identity.

But it’s also true that If you did L first,then M,

you’d also get the identity.

So since L undoes M and is undone by M,

L and M are inverses of each other.

[L和M互为对方的逆！]
You might wonder, is the left inverse

always also the inverse like this?

No, obviously not, or they wouldn’t have different names, would they?

[不能剧透！！！]
Before we move on, let me ask you a question

to check you’ve understood this so far.

Imagine you have a matrix like this.

What it does is that it takes a 3D vector, and jumbles up the components.

Does this matrix have a left inverse?

As in, can you undo this?

Then if it does have a left inverse,

figure out if it has an inverse as well.

Put you answer in the poll in the corner,

and pause the video now to think about it.

[A矩阵有左逆吗？ A矩阵有逆吗？]
The answer is that it does have a left inverse.

It’s the one that takes a vector like this

and rearranges the components like this.

It’s clear that this is a left inverse of A

since it undoes it like so.

But A is also a left inverse of it,

as you can see, because A undoes this matrix.

So B is the inverse of A.

Now that we know what an inverse is,

let’s think more about when they exist or don’t.

Again, it’s going to be more convenient to look at

when a left inverse exists first.

Here’s another question.

This matrix takes a 2d vector a b and sends it to a 0.

Does this matrix have a left inverse?

If so, figure out what it is.

in the poll in the corner and pause now to think about it.

It takes the vector a b to a 0,

but it also takes a d to a 0 as well.

and it’s because of this that the left inverse doesn’t exist. Why?

Well, say you have some vector V and M takes it to W.

The left inverse of M, if it exists, knows M and what W is,

but it doesn’t know what the vector that produced W was.

Just using the information given, it needs to find what the original vector was,

so that it can take W back to where it came from.

However. If there’s some other

vector U that also goes to W,

the left inverse has a problem.

it can’t just look at W and know for sure it whether

it came from V or from U because there isn’t enough information.

This means the left inverse

*can’t* take w back to where it came from, so… it doesn’t exist.
“不能”对W进行还原 所以左逆不存在
This thing here,

where two different vectors V and U

get mapped to the same vector,

i.e M(U)=M(V),

is what I’ll call M losing information.

What we’ve just seen is that if M loses information

it doesn’t have a left inverse.

But what about the other way around?

[M矩阵无信息丢失]
If M doesn’t lose information

does this mean the left inverse exists?

Well, yes actually.

Because all the left inverse has to do

to undo M is find the vector W came from.

Since there’s only one vector V it could be,

there is an inverse that takes W and returns V.

This doesn’t mean it’s easy to find out what V is necessarily,

but looking at W does in principle give you enough information

to undo M and return V.

So a matrix has a left inverse if and only if

it doesn’t lose information.

[信息不丢矢 左逆有意义]
Let’s look at another example

to understand this point better.

Imagine I have a matrix from 2d to 3d

and what it does is,

it rotates any 2D vector into 3D space like this.

[不好意思，再来一次]
[我真是个小机灵]
Does this matrix have a left inverse?

Pause the video to think about it.

The answer is, it does have a left inverse

because A doesn’t lose information.

If you want to take vectors like this back,

you know where they came from

so all you have to do is rotate the plane back.

Let B be a matrix that

takes 3D vectors to 2D that does rotate this plane back.

It is a left inverse of A.

Now, is B the inverse of A?

In other words, is A B’s left inverse?

Pause the video and think about it.

The answer is, no, B has no left inverse

We’ll show that by showing B loses information.

First, pick any 3D vector that’s not on this plane

[不是这个面]
B has to send it to some 2D vector,
B矩阵要将它变换成二维向量
so let’s just say it’s here.

But there’s another 3D vector that’s been already sent there.

It’s this vector u that’s on the plane.

So B(u) is equal to B(v).

Hence B loses information and doesn’t have a left inverse.

There’s an important lesson to be drawn from this example.

You might have wondered

why we only ever talk about the inverses of a square matrices.

What’s so special about transformations from n dimensions to n dimensions?

The reason is, non square matrices, i.e.

ones from n dimension to m dimensions never have an inverse.
n维到m维的变换矩阵永远不会有逆矩阵
The issue is, if you have any transformation going from a bigger space to a smaller space,

like B which went from 3d to 2d,

you have to lose information.

These types of matrices always send some vectors

to the same place.

If you have a matrix from a smaller to a bigger dimension,

it’s possible that it has a left inverse

like A did in our example.

But it’s left inverse goes from big to small, like B,

and so it can’t be undone.

Hence, even though some nonsquare matrices have left inverses,

they never have an inverse.

Square matrices don’t have these issues at all though.

Actually, for square matrices, everything massively simplifies

because the left inverse is always equal to the inverse.

So if a square matrix has a left inverse,

it automatically has an inverse.

I am not going to prove this fact.

You are, for homework.

But I will give you an illustrative exanple in a little

[对于方阵 左逆即逆 我叫它“例证法” 但我的线代老师却不屑一顾]
Once you’ve proved it, you’ ll see that for a square matrix A,

if B undoes A,

then A undoes B as well.

This gives us an easy criteria

for checking whether A has an inverse or not,

because it’s the same criteria we used to check

whether A has a left inverse:
A是否有左逆矩阵的办法相同：
Just ask, does A lose information.

If yes… then sorry, A inverse doesn’t exist.

If no, then A inverse does exist.

You might be thinking, ok, so what?

How is this easy to check?

Wouldn’t you have to compute the outcome for every single vector

that goes into M and compare the results to every other vector

and see if any of them match?

Isn’t that beyond tedious?

Thankfully there is an easy way to check this condition.

All you have to do is figure out

which vectors get mapped to 0.

For any linear map,

0 is always mapped to 0,

but all you need to do is find out

if there’s any *other* vector mapped to 0 or not,

and that’s enough to decide if M loses information.

Why would that be enough?

Imagine two vectors U and W do both go to W,

so M loses information.

Then the vector U-V, by linearity, gets mapped to 0

So whenever you have 2 vectors going to the same thing like this,

you always get at least 2 vectors ending up at 0.

So you can check whether a matrix loses information

by just checking how many things go to zero.

In other words, figure out how many vectors V

satisfy the equation MV=0.

This is part of why you spend so much time

in a linear algebra course studying the solutions to equations like this.

You can solve for V by a) using subsitution

b) using Gaussian elimination

or (c) by getting your computer to do the Gaussian elimination for you.

But the point is, you can find out if M loses information easily enough this way

and that tells you whether M has an inverse.

Let me summarize quickly

what we learnt in this video:

[因为这个视频有点儿长 既对你们 也对我们]
1. A left inverse of a matrix is matrix that undoes it.

2. That the left inverse exists if you don’t lose information:

i.e, if the matrix never sends two different vectors to the same vector.

3. The inverse is the matrix that both

undoes the matrix, and is undone by the matrix

4. Nonsquare matrices never have inverses.

5.For a square matrix,

the left inverse is equal to the inverse

6. You only need to check if a square matrix

loses information or not to decide if it has an inverse.

7. You can check whether the matrix loses

information by looking at how many different vectors get mapped to 0.

This you can do by solving the equation Mv=0.

And so that’s it. But before you run off,

here’s some homework for you.

The first one is multiple choice.

Which of these is the inverse of this matrix?

I know that you can just check

each of these to see which works,

but I’d rather you did it another way.

[就当这种取巧的方法配不上俺们这些好学生吧]
And for crying out loud,

don’t use the formula for the inverse.

Question 2. Prove that for a square matrix

M inverse is equal to the left inverse of M.

There’s lots of hints in the description for this,

but first I want you to try question 3

because it’s a very illustrative example.

[当我说我来做例题演示的时候 其实是要你来做：）]
Question 3. Suppose we have the matrix from before.

If we have a left inverse for it, L,

then we know L undoes M.

We also know M takes the basis vectors to these vectors,

so L must take them back.

First show that these two new vectors form a basis.

Then, to show that M undoes L,

we need to show that for any vector v, if you apply L

and then M, you get v back.

Show this by writing v as

a linear combination of the basis in the first part.

Hopefully doing this first will help you with the proof in question 2.

As you will have noticed, the first question was from Brilliant.org.

What I like about questions from there it is

that they don’t just give you a formula and then ask questions

where you plug numbers into that formula.

That’s what I found a lot of highschool and early university textbooks do,

and it’s annoying because that doesn’t teach you anything.

Instead, they get you to do examples like this one,

and then understand the principle yourself,

then in the next few questions,

As you can see,

this is exactly how I like to learn new maths

before reading the proof of anything I’ll do lots and lots of simple examples first

and try to understand the underlying reasoning

so I love how Brilliant allows you to do this in a structured way.

They have loads of different maths and science course on their website,

which you can access completely with a monthly or yearly membership.

or on screen, you can get 20% off an annual subscription.

Alright, so that’s all for this video,

I hope you enjoyed it.

The next one is about changing basis,

which is key to understanding loads of Quantum mechanics,

including the Heisenberg uncertainty principle.

It should be up here in about 2 weeks,

as long as I don’t dislike I hate it just after uploading it

like I did with the original version of this one.

Anyway, subscribe if you’d like to be notified

when my next one is up. Thanks for watching!