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Kreisausschnitt berechnen (Sektor) - einfach erklärt von Lehrerschmidt

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Mathe

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Algebra

Erklären

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erklärt

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Intuition

Beispiel

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Idee

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vektorraum

00:00:06

yes hello and a very warm

00:00:07

welcome to MEF Intuition, I am

00:00:09

Markus and in this video I would like to

00:00:11

tell you something about the dual space, which

00:00:14

I think is a very difficult

00:00:15

concept to imagine. You will

00:00:17

see why later, but

00:00:19

above all it should it's about how

00:00:21

can you imagine it? I want to

00:00:22

give you a good idea of

00:00:25

what you have to imagine by it, what will

00:00:26

help you imagine it, we

00:00:28

'll look at the definition

00:00:30

and two examples and

00:00:33

I'd like to start with the motivation for what you

00:00:35

need You can do the whole thing at all, yes, and

00:00:38

I'll go into it a little bit and

00:00:40

imagine the following: imagine you

00:00:42

have a pane of glass, yes, this is

00:00:45

your level and, alternatively, you have

00:00:49

something that looks like a

00:00:51

tablecloth that hangs down over the edge of the table

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yes, it's also like

00:00:56

something two-dimensional that

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somehow hangs down here and I want to

00:01:00

examine these two objects. Yes, that

00:01:02

's X1 and that's I would

00:01:10

really just like

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to give you an idea and I would like to do the following:

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I would like to draw piles for each of these two

00:01:17

objects that are

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somehow exactly perpendicular to their surface

00:01:22

and then I get

00:01:25

a picture up here and I can

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do it again down here and

00:01:30

here on the side something like this is created.

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Yes, this is what I got from it.

00:01:36

Now imagine that you have a

00:01:39

highly abstract object x1 and x2 that you

00:01:42

can't draw so easily,

00:01:44

but you are also able to do so, I'll tell you

00:01:47

quite formally To draw such piles

00:01:49

and examine them quite easily,

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yes and then you could, for example,

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notice that the arrows at X1 are

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all parallel to each other, while at

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First of all, they are all parallel,

00:02:08

but if I take one from above and below,

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then of course it goes wrong

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and if I were blind, so to speak,

00:02:14

to x1 and x2 as I

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imagine them, then this

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knowledge of how the piles are related to each other helps

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me It's better to imagine X1 or the

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I know that

00:02:40

I don't have a pane of glass like that, there

00:02:42

's a curvature in it somewhere

00:02:44

because the arrows aren't all

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parallel, so I do

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n't need to go too much into physics, but

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as I said, it's just a

00:02:51

small idea and something We have

00:02:53

now created it, we have

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managed it, we wanted to examine our x,

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i.e. either X1 or

00:03:10

Quantity works,

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yes, math always consists of quantities and

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quantities are then the objects that you

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try to understand, yes and

00:03:19

I just say it's a very cool trick that

00:03:21

someone found at some point that

00:03:23

he said I'll take a look at when I'm talking

00:03:25

about mine Object that I don't yet

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understand very well when I

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look at an illustration in some other

00:03:31

set and then see if I can understand these

00:03:32

illustrations

00:03:34

better,

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yes that's the idea

00:03:38

behind it and here we come namely

00:03:40

to define the

00:03:42

dual space and I always want to define the

00:03:45

dual space for a vector space

00:03:47

and that is this vector space

00:03:49

which now plays the role of the

00:03:55

a K

00:03:59

vector space yes this K is

00:04:01

always part of it we make the whole thing

00:04:03

clear here we have the R

00:04:05

square as an example and the R that

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I have then what is already in here

00:04:09

is my body yes that is

00:04:11

like mine Axis yes, you can

00:04:13

also draw it in your head yes

00:04:15

in this case it is actually something like

00:04:17

this pane of glass that is my V my R

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square and this axis of this R that

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is already in here twice in one place

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yes okay and I

00:04:28

define this dual space I now as

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a certain set yes and

00:04:35

this set has one of two symbols in

00:04:38

your lecture you will either

00:04:39

find a V line or you will find

00:04:42

a V star and that is then defined

00:04:45

as a set that consists of

00:04:48

maps F from V to K With the

00:04:54

property like this you always read this

00:04:55

line that my figure F

00:04:58

is linear, that's

00:05:01

not so important for the view at this point

00:05:02

to understand why it has to be that way,

00:05:05

um that actually just ensures

00:05:07

that what's here now The result is that it

00:05:09

also has a certain structure because

00:05:11

being linear as a mapping is

00:05:13

simply another term for

00:05:15

saying that it is a homomorphism that is

00:05:17

present here and this homomorphism

00:05:20

is a structure-preserving

00:05:22

mapping. I have already

00:05:24

made a video about it It's

00:05:25

not supposed to work here now, but what you're

00:05:27

imagining is that you want to understand this V

00:05:31

better and define

00:05:34

a set of partners, so to speak, that's why

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there's this V in it again,

00:05:37

just look at it, yes, and now that's

00:05:41

a set whose elements

00:05:44

are illustrations and you've just learned why it

00:05:46

makes sense to look at the illustration

00:05:48

because you want to better understand this object

00:05:50

here in front where the illustration starts

00:05:52

and that's what we're going to do

00:05:54

now so we're going to draw here

00:05:55

what comes out here

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first of all I have here so my k k is

00:06:01

in In the case, the set of real

00:06:03

numbers and now I have

00:06:05

illustrations or I have an

00:06:07

illustration which

00:06:09

I will now start with. So I will take

00:06:10

an illustration which

00:06:12

always gets a general element from the

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left side namely some vector

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yes and a vector here is

00:06:18

just like a point in

00:06:19

my plane which is given by two

00:06:21

coordinates X and Y and my illustration

00:06:24

sends it somewhere and what

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comes out of it should be an element in

00:06:28

my K and in In our case

00:06:31

these are the real numbers and it

00:06:33

also has to be a linear mapping and

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that's why I'll give you a few

00:06:37

examples of linear mapping e.g. X +

00:06:41

y or what you could also do would be to

00:06:44

write factors here in front of x or

00:06:46

in front of y a times

00:06:59

That would

00:07:01

all be a linear mapping, but it

00:07:03

wouldn't be a linear mapping if it said

00:07:05

x times y, okay,

00:07:07

but it's not that important at this point, and

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what's important for you to understand this concept of

00:07:12

dual space is the

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following: the elements in here are

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like said illustrations yes, that means

00:07:20

this whole picture that you see here

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yes you have a left side you then have

00:07:24

a specific illustration of how an element

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from the left side is assigned to

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a real number and you do that

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for each element here on the On

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the left side you can now

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imagine here like an arrow from left

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to right, yes for every point in the

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plane you get a PIL that goes somewhere

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here

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and the whole picture here, this

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whole figure is only a

00:07:46

single element in your dual space, okay

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so and now comes the crazy thing: you have

00:07:53

your vector space V and your vector is a

00:07:56

vector space if you have one like that consists

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of vectors yes you simply call the

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elements vectors if they

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are part of a vector space and the

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crazy thing is now this dual space

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that can do it again become a vector space,

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yes you

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only have to do two things for that because a

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vector space consists of the elements

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and you can simply add them somehow,

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just like you could add the vectors here

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in the plane and you

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have to stretch or compress vectors with a

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scalar multiplication

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These are the two

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conditions for a vector space and this

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is still formally

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defined somewhere and a few properties have to

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apply, but that's

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not important at all. What's important for you is

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this vector space, um, which is now

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created here, this dual space which is

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Now, of course, it's very difficult to imagine,

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yes, because you already started with a

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vector space like this. You

00:08:53

might have been able to imagine it

00:08:54

as a plane, but now it's driving me

00:08:56

crazy because what

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is an illustration here is an element in

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this vector space and is therefore

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something like a vector and

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I think that's the big

00:09:08

difficulty in imagining this term

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because what does it mean when

00:09:11

an illustration is a vector, yes and

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that's why the memory is

00:09:15

a vector space m and that's where it goes now

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The formality only consists of this

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set and you should be able to add the elements in it

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somehow.

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Then there has to be a rule as to how

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you combine these elements with each other and

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there has to be a scalar multiplication.

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Yes, those are the two

00:09:35

conditions for the

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vector space, roughly speaking Yes, and now the question is

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how can I add figures

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or with a scalar multiplication like

00:09:45

stretching and compressing, you ca

00:09:47

n't imagine that anymore, but

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I'll tell you the formality behind it, it

00:09:51

's actually quite simple, you

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actually already know how to

00:09:55

add figures and multiply and

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I'll do an example of F from x= x

00:10:02

and G from x = x²- 2, these are both

00:10:07

maps, yes you can now

00:10:08

imagine that they are maps from the

00:10:09

real numbers into the real numbers

00:10:11

or something else, yes that's it now

00:10:12

Not so important here, but you know from

00:10:16

school how you could add something like that

00:10:18

by writing

00:10:20

x² + x- 2 yes and what

00:10:25

happened? You simply trace the

00:10:28

definition of addition back

00:10:31

to an addition that you already know,

00:10:33

namely in a body there you

00:10:36

already know how you can add or

00:10:38

multiply in our

00:10:40

example these are the real numbers you

00:10:41

have known since school how you can

00:10:43

add and multiply in the real numbers

00:10:45

yes and that will help

00:10:47

you here too because you

00:10:50

just define I take it

00:10:53

Now here's two illustrations, one I call

00:10:54

f and the other I call g and I

00:10:56

want to add them together now what comes

00:10:59

out of that and I define the

00:11:03

illustration by what it of course

00:11:04

does with an element which it gets from V

00:11:06

yes so I now define what

00:11:09

pass what is the sum of f + g of V

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yes because what comes out here should

00:11:15

be a figure again and it

00:11:17

gets an element V yes and I

00:11:19

simply define it as F of V + g of

00:11:25

V so and that what Here it is

00:11:29

of course an element from K yes because we

00:11:32

know this F that should be an

00:11:34

element here from my dual space,

00:11:36

that means if I have applied the F to an element

00:11:38

then I am in an

00:11:40

element from my body yes that is exactly how

00:11:42

I am here over there in a body and the

00:11:44

crazy thing is that it's a body,

00:11:45

I already know how to

00:11:47

add in it, yes, that's the

00:11:49

body addition that I

00:11:51

use here, yes and that's why it creates

00:11:54

something exactly like the polynomial

00:11:56

down here and exactly the same Can I also add an

00:11:58

element an image like stretch

00:12:00

I take a body element K and

00:12:03

I take my image yes I

00:12:05

take an image from here that

00:12:07

I want to stretch yes my F

00:12:08

I want to stretch and now of course I have to

00:12:10

know this stretched

00:12:12

image yes it then gets the symbol

00:12:14

K times F what does it do with my V

00:12:18

yes and that is then only

00:12:19

defined as a multiplication that

00:12:22

I already have in K yes because I can

00:12:24

of course multiply an element K from my body

00:12:27

by another element

00:12:28

what is already in my body and

00:12:30

this F of V is of course

00:12:32

an element from K, which means

00:12:34

here I have the multiplication of

00:12:36

my body K which I repeat here, yes, but

00:12:39

I'll get rid of that again and

00:12:41

that's why I can elements in here

00:12:46

Add and multiply in, I would say, an obvious way even though they

00:12:49

are illustrations, yes, and that is exactly

00:12:52

the difficult part that

00:12:54

I think is explained far too rarely at this point

00:12:56

and yes, the idea behind it

00:12:59

is, as I said, a dual space is better for this because

00:13:01

the original vector space is better

00:13:04

examine, by the

00:13:06

way, there is also a very,

00:13:09

very nice geometric interpretation

00:13:12

of it, which I would also like to

00:13:13

explain to you, it's really great, you can

00:13:15

impress your friends with it, and with

00:13:17

Platonic solids

00:13:21

there are five of them, yes the one is m but I

00:13:26

call it now just a few and

00:13:27

show you what m so one

00:13:29

is the

00:13:31

tetrahedron, the other is the

00:13:34

cube and then we have the

00:13:36

octahedron and then I'll write them down now but I

00:13:38

won't add them, we have an

00:13:40

ikosaeda and a dodecaeda yes these

00:13:43

You can draw it quite nicely

00:13:44

here, it

00:13:46

's something like that

00:13:48

here, that means wait exactly, yes, it has

00:13:51

four faces, it's all triangles. Of

00:13:53

course, you also know the cube

00:13:56

here and the octahedron, which looks

00:14:01

similar to the tet with

00:14:03

triangles, however You don't just have

00:14:04

four pieces of it, but you have something

00:14:06

like a pyramid, two pieces that

00:14:08

are on top of each other and then the whole thing looks like this,

00:14:14

okay, and now let's do the

00:14:18

following: we take the cube and

00:14:20

do the following operation in

00:14:22

front of you this cube here this

00:14:24

object would now be called V yes and now

00:14:26

there is something like this you can

00:14:28

transfer this cube object into another

00:14:30

object and I will call that before

00:14:32

yes and I would like to do the following

00:14:35

I always take the center of

00:14:37

one side

00:14:38

here and then I connect them all

00:14:41

together, something like this comes out of it. There's

00:14:43

also one missing at the back,

00:14:47

an octahedron comes out. Yes,

00:14:51

the picture we have here, I can do the same thing,

00:14:54

but when I

00:14:55

do the okaeda, the same game

00:14:57

again, I take the center points here Here

00:14:59

that always means I have four at the bottom and

00:15:02

I have four at the top and I connect them

00:15:04

together and what comes out of it, you guessed

00:15:07

it, the cube is found here again

00:15:10

when I do the whole thing for the tetrahedron,

00:15:12

I take the center points here,

00:15:15

what comes out of that and maybe

00:15:17

You'll soon recognize it, so now it's

00:15:19

obviously very small, it's

00:15:20

a tetrahedron again, yes, that means what

00:15:23

we discovered here happened at this

00:15:24

point. V is the same as V

00:15:27

asterisk and here we saw

00:15:31

V becomes V asterisk of the cube becomes

00:15:34

an octahedron through this operation and

00:15:37

the octahedron becomes a cube and

00:15:41

of course pay attention now that we are back

00:15:43

to vector spaces there is a very,

00:15:45

very interesting theorem and this theorem

00:15:47

which says that

00:15:49

under certain conditions it is true that V

00:15:52

is the same as if I take the dual space

00:15:54

and, viewed on its own,

00:15:57

form the space again because,

00:15:59

as I said, this dual space is a

00:16:01

vector space and

00:16:03

no one forbids me from trying to imagine the dual space for this vector space,

00:16:07

yes, and that's what it's called, by

00:16:09

the way the bidual space because I

00:16:11

just do it twice, yes, and this sentence

00:16:13

that says if my V

00:16:15

is finitely dimensional as

00:16:18

a vector space then these two

00:16:20

objects are almost equal, they say

00:16:22

isomorphic, so you can imagine

00:16:24

something like an equal sign in

00:16:27

quotation marks, yes, so it

00:16:29

is isomorphic the formally correct term, which is

00:16:31

basically the geometric idea

00:16:34

of it, where you can also find duality in

00:16:37

geometry. Yes, these are

00:16:39

simply objects that

00:16:41

are related to each other in some way. Yes, you can

00:16:42

then understand the cube better

00:16:45

if you understand the octahedron better, yes

00:16:47

or not On the other hand, if you

00:16:50

want to understand the octahedron then you can

00:16:51

use the cube yes and

00:16:54

yes that's just a nice idea. I only

00:16:58

learned this point quite late in my studies. I found it

00:16:59

pretty cool, um exactly. I hope the

00:17:02

video helped you a little, yes

00:17:04

and if you have any more questions about the

00:17:06

topics from the algebra line then

00:17:08

I can highly recommend my course in

00:17:10

this direction,

00:17:11

you can find the link to it on my website here

00:17:13

or you can

00:17:16

also write to me in general terms there If you

00:17:19

have any questions or video requests, yes and otherwise, see you in

00:17:21

the next video

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The best quality formats are FullHD (1080p), 2K (1440p), 4K (2160p) and 8K (4320p). The higher the resolution of your screen, the higher the video quality should be. However, there are other factors to consider: download speed, amount of free space, and device performance during playback.

Why does my computer freeze when loading a "Dualraum - intuitiv erklärt! | Math Intuition" video?

The browser/computer should not freeze completely! If this happens, please report it with a link to the video. Sometimes videos cannot be downloaded directly in a suitable format, so we have added the ability to convert the file to the desired format. In some cases, this process may actively use computer resources.

How can I download "Dualraum - intuitiv erklärt! | Math Intuition" video to my phone?

You can download a video to your smartphone using the website or the PWA application UDL Lite. It is also possible to send a download link via QR code using the UDL Helper extension.

How can I download an audio track (music) to MP3 "Dualraum - intuitiv erklärt! | Math Intuition"?

The most convenient way is to use the UDL Client program, which supports converting video to MP3 format. In some cases, MP3 can also be downloaded through the UDL Helper extension.

How can I save a frame from a video "Dualraum - intuitiv erklärt! | Math Intuition"?

This feature is available in the UDL Helper extension. Make sure that "Show the video snapshot button" is checked in the settings. A camera icon should appear in the lower right corner of the player to the left of the "Settings" icon. When you click on it, the current frame from the video will be saved to your computer in JPEG format.

What's the price of all this stuff?

It costs nothing. Our services are absolutely free for all users. There are no PRO subscriptions, no restrictions on the number or maximum length of downloaded videos.