
Korean: 
여러분이 행렬 C가 있다고 가정합시다
여러분이 행렬이라는 것을 더 이해시키기 위해
C를 진하게 쓰겠습니다
그리고 행렬 C는 행 a와 b열로 이루어져 있습니다
그리고 행렬 C는 행 a와 b열로 이루어져 있습니다
이 행렬은 a×b 행렬입니다
그리고 이 행렬에
단위 행렬을 곱해보겠습니다
한 번 더 행렬이라는 것을 나타내기 위해
문자를 진하게 쓰겠습니다
우리는 단위 행렬 I에 행렬 C를 곱할 것이고
이 값은 당연히 행렬 C일 것입니다
왜냐하면 저 부분은 단위 행렬의 영역이고,
왜냐하면 저 부분은 단위 행렬의 영역이고,
C는, 이미 알고 있듯이 a×b행렬입니다
A행과 B열의 행렬입니다
그렇다면 이에 기초하여, I의 차수는
어떻게 됩니까?

English: 
Voiceover:Let's say that
you've got some matrix C,
trying my best to bold it,
to make sure you realize
that this is a matrix.
Let's say that we know that it has a rows
and b columns.
It's an a by b matrix.
Let's say that we are going to multiply it
by some identity matrix.
Once again let me do my
best to attempt to bold this
right over here.
We're going to multiply the
identity matrix I times C
and of course we are going to get C again
because that's the identity matrix,
that's the property of
the identity matrix.
Of course C, we already
know is an a by b matrix.
A rows and b columns.
Based on this, what
are the dimensions of I
going to be?

Czech: 
Řekněme, že máte nějakou matici C.
Pokusím se ji co nejlépe zvýraznit,
abyste si snadno představili,
že se jedná o matici.
Řekněme, že víme, že má
"a" řádků
a "b" sloupců.
Je to tedy matice typu axb.
Řekněme dále, že tuto matici vynásobíme
nějakou jednotkovou maticí.
Nyní se opět pokusím ji vyznačit
co možná nejtučněji.
Provedeme součin jednotkové
matice I krát matice C
a samozřejmě pak opět dostaneme matici C,
protože se zde jedná o jednotkovou matici
a ta má právě takovou vlastnost.
Jistě už víme, že matice C je typu axb.
Má "a" řádků a "b" sloupců.
Pokud z tohoto budeme vycházet,
jakého typu bude matice I?

Thai: 
สมมุติว่าคุณได้เมทริกซ์ C
พยายามทำตัวหนาให้ดีที่สุดนะ
เพื่อให้แน่ใจว่าคุณเห็นเป็นเมทริกซ์
สมมุติว่าเรารู้ว่ามันมี a แถว
และ b คอลัมน์
มันคือเมทริกซ์ a คูณ b
สมมุติว่าเราจะคูณมัน
ด้วยเมทริกซ์สักตัวหนึ่ง
เหมือนเดิม ขอผมพยายามทำตัวหนา
อันนี้หน่อย
เราจะคูณเมทริกซ์เอกลักษณ์ I คูณ C
และแน่นอน เราจะได้ C เหมือนเดิม
เพราะนั่นคือเมทริกซ์เอกลักษณ์
นั่นคือสมบัติของเมทริกซ์เอกลักษณ์
แน่นอน C เรารู้แล้วว่ามันคือเมทริกซ์ a คูณ b
a แถวกับ b คอลัมน์
จากตรงนี้ มิติของ I
จะเป็นเท่าใด?

Bulgarian: 
Да кажем, че имаш
някаква матрица C...
Старая се да го удебеля,
за да сме сигурни, че
това е матрица.
Да кажем, че знаем, че 
тя има a броя редове
и b броя колони.
Това е матрица a x b.
Да кажем, че 
ще я умножаваме по
някаква единична матрица.
Пак ще се опитам 
да го удебеля колкото мога.
Ще умножим 
единичната матрица I по C
и естествено 
ще получим пак C,
защото това
 е единична матрица
и тя има такова свойство.
Вече знаем, че C 
е матрица a х b.
а броя редове и 
b броя колони.
Като знаем това, какви мислиш, 
че ще са размерите на I?

Thai: 
ผมแนะนำให้คุณหยุดวิดีโอนี้ และคิด
ด้วยตนเอง
เราได้ทำแบบฝึกหัดนี้มาแล้วนิดหน่อย
โดยเราดูเมทริกซ์เอกลักษณ์
แต่ตอนนี้เราจะทำ --
เราจะคูณเมทริกซ์เอกลักษณ์
ด้วยเมทริกซ์ทั่วไปมาก
ผมแค่พูดโดยทั่วไป
เกี่ยวกับมิตินี้
สิ่งหนึ่งที่เรารู้คือว่าการคูณเมทริกซ์
จะนิยาม
เมื่อคอลัมน์ จำนวนคอลัมน์
ของเมทริกซ์แรกเท่ากับจำนวนแถว
ของเมทริกซ์ที่สอง
อันนี้มี a แถว อันนี้จึงมี a คอลัมน์
ทีนี้ อันนี้จะมีกี่แถว?
เรารู้แล้วว่าการคูณเมทริกซ์นิยามได้
ถ้าจำนวนคอลัมน์ในเมทริกซ์แรก
จะเท่ากับจำนวนแถวในคอลัมน์ที่สอง
เรารู้ว่าผลคูณจะได้จำนวนแถวของมัน
จากจำนวนแถว
ของเมทริกซ์ตัวแรกที่จะคูณ

English: 
I encourage you to pause
this video and think about it
on your own.
We've already done this
exercise a little bit,
where we first looked at identity matrices
but now we're doing it with a very ...
We're multiplying the identity matrix
times a very general matrix.
I'm just even speaking in generalities
about these dimensions.
Well one thing we know is
that matrix multiplication
is only defined
is if the column, the number of columns
of the first matrix is
equal to the number of rows
of the second matrix.
This one has a rows, so this
one's going to have a columns.
Now how many rows is
this one going to have?
We already know that matrix
multiplication is only defined
if the number of columns
on the first matrix
is equal to the number of
rows on the second one.
We know that the product
gets its number of rows
from the number of rows
of the first matrix being multiplied.

Bulgarian: 
Насърчавам те да спреш видеото
и да помислиш самостоятелно.
Вече сме правили
това упражнение,
когато разглеждахме
единичните матрици за първи път,
но сега умножаваме
единичната матрица
по много обща матрица.
Даже говоря с общи термини
за размерите.
Едно нещо, което знаем, е, че умножението
на матрици е дефинирано
само когато
броят на колоните
от първата матрица е равен
на броя на редовете
от втората матрица.
Тази има а броя редове, следователно
тази ще има а броя колони.
Колко реда 
ще има тази?
Вече знаем, че умножението на
матрици е дефинирано само когато
броят на колоните 
от първата матрица
е равен на броя на 
редовете от втората.
Знаем, че броят на редовете
от произведението се получава
от броя на редовете
от първата матрица.

Korean: 
여러분께서는 잠시 영상을 멈추시고
직접 풀어보시기 바랍니다
우린 이미 항등 행렬 부분에서
연습을 하였습니다, 하지만
이번에는 항등 행렬과 일반적인 행렬을
이번에는 단위 행렬과 일반적인 행렬을
곱해야 합니다
심지어 저는 차수의 일반성을
얘기하고 있었습니다
우리가 알아야 할 것은 행렬의 곱셈이
첫 번째 행렬의 열 개수가 두 번째 행렬의
행의 개수와 같을 때에만
적용된다는 것입니다
적용된다는 것입니다
이 부분은 행을 가지고 있으므로 
이 행렬은 열을 가졌을 것입니다
그렇다면 이 행렬의 행의 개수는 얼마입니까?
우리는 이미 행렬의 곱셈이
첫 번째 행렬의 열의 개수와 두 번째 행렬의 행의 개수가
같을 때만 적용된다는 사실을 알고 잇습니다
그리고 곱해진 값은
첫 번째 행렬의 행의 개수가 곱해진 값을
받게 됩니다

Czech: 
Nyní by bylo dobré si zastavit video
a zamyslet se nad tím
na vlastní pěst.
Příklad podobného typu
jsme již dříve dělali,
a to když jsme si poprvé
uvedli jednotkovou matici,
ale nyní se jedná o velmi…
Násobíme jednotkovou matici
jinou zcela obecnou maticí.
Pracuji zde s naprostou obecností,
co se týče typu této matice.
Víme, že násobení matic
je definováno jen tehdy,
pokud počet sloupců první matice
je roven počtu řádků druhé matice
Tato matice má "a" řádků, tedy
tahle bude mít "a" sloupců.
Kolik řádků pak bude mít tato matice?
Již víme, že násobení matic
je definováno jen tehdy,
pokud počet sloupců první matice
je roven počtu řádků druhé matice.
A víme, že výsledek součinu
bude mít počet řádků
daný počtem řádků první matice ze součinu.

Czech: 
Výsledek součinu pak má "a" řádků,
tedy jednotková matice pak
musí mít "a" řádků.
A co je na tom zajímavé?
Když jsme se poprvé seznámili
s jednotkovými maticemi,
které jsme dále násobili,
pracovali jsme s příkladem
s maticemi typu 3x3
a ve výsledku jsme dostali
jednotkovou matici typu 3x3.
Co jsme si zde odvodili je zajímavé v tom,
že jednotková matice je pro
libovolnou matici,
dokonce i pro ne-čtvercovou matici,
kde "a" a "b" mohou být
dvě různá čísla.
Pak jednotková matice
k libovolné matici
bude čtvercovou maticí.
Bude mít stejný počet řádků
jako bude její počet sloupců.
Když se zamyslíme nad
jednotkovými maticemi,
můžeme se vskutku bavit ve stylu…
je tato jednotková matice
typu 4x4?
Je typu 3x3?
Je typu 2x2?
Nebo možná 1x1?
Zvykem pak je, že nepíšeme,
že jednotková matice typu 2x2

Bulgarian: 
Щом произведението има
а броя редове,
значи и единичната
матрица тук
ще има а броя редове.
Какво е интересното
при това?
Когато първо се запознахме
с единичните матрици
и умножавахме,
избрахме пример
с размери 3 x 3
и единична матрица 3 х 3.
Интересното нещо, което
сега си доказахме,
е че единичната матрица 
за всяка матрица,
дори да не е квадратна,
а и b могат да бъдат 
с различни стойности...
Единичната матрица 
за произволна матрица
ще бъде квадратна матрица.
Ще има еднакъв брой
редове и колони.
Когато мислим за
единични матрици,
можем просто да се питаме: 
Тази единична матрица
4 х 4 ли е?
3 х 3 ли е?
2 х 2 ли е?
Или пък 1 х 1?
Подходът даже не е да пишем, 
че единичната матрица

English: 
The product has a rows
then the identity matrix right over here
has to have a rows.
What's interesting about this?
When we first got introduced
to identity matrices,
we were multiplying,
we picked out a three by three example
and we got a three by
three identity matrix.
What's interesting about what
we've just proven to ourselves
is the identity matrix for any matrix,
even a non square matrix,
a and b could be two different values.
The identity matrix for any matrix
is going to be a square matrix.
It's going to have the same number of rows
and the same number of columns.
When we think about identity matrices,
we can really just say, well
is this the identity matrix
that is a four by four?
Is it a three by three?
Is it a two by two?
Or I guess one by one?
The convention is, it isn't
even to write identity

Korean: 
곱셈값은 행이 있고
이 말은 단위 행렬에 행이 존재한다는 것입니다
이 말은 항등행렬에 행이 존재한다는 것입니다
여기서 신기한 사실은
우리가 항등 행렬을 처음 접했을 때
우리는 곱셈을,
우리는 3×3 행렬을 예로 들었고
3×3 단위 행렬을 받았습니다
여기서 신기한 것은 우리가 증명한 것입니다
그 어떤 행렬의 단위 행렬에서
그 어떤 행렬의 정방 행렬에서
행과 열의 값이 다를 수도 있다는 것입니다
임의의 행렬의 단위 행렬은
정방 행렬이 될 것입니다
그 행렬은 같은 수의
행과 열이 있을 것입니다
우리가 항등 행렬에 대하여 생각할 때
우리는 그냥 단위 행렬이
4×4이나
3×3이거나
2×2이거나
1×1 행렬이라고 추측할 것입니다
관례상 항등 행렬 밑에

Thai: 
ผลคูณมี a แถว
แล้วเมทริกซ์เอกลักษณ์ตรงนี้
ต้องมี a แถว
สิ่งที่น่าสนใจในนี้คืออะไร?
ตอนเราเรียนเรื่องเมทริกซ์เอกลักษณ์
เราจะคูณ
เราเลือกตัวอย่างขนาด 3 คูณ 3
และเราได้เมทริกซ์เอกลักษณ์ 3 คูณ 3
สิ่งที่น่าสนใจ ตอนที่เราพิสูจน์ด้วยตนเอง
คือเมทริกซ์เอกลักษณ์สำหรับเมทริกซ์ใดๆ
แม้แต่เมทริกซ์ที่ไม่ใช่เมทริกซ์จัตุรัส
a กับ b มีค่าต่างกันสองค่าได้
เมทริกซ์เอกลักษณ์สำหรับเมทริกซ์ใดๆ
จะเท่ากับเมทริกซ์จัตุรัส
มันจะมีจำนวนแถวเท่ากัน
และจำนวนคอลัมน์เท่ากัน
เมื่อเราคิดถึงเมทริกซ์เอกลักษณ์
เราบอกได้แค่ว่า เมทริกซ์เอกลักษณ์นี่เป็น
4 คูณ 4 ได้ไหม?
มันเป็น 3 คูณ 3 ได้ไหม?
มันเป็น 2 คูณ 2 หรือเปล่า?
หรือ 1 คูณ 1?
ธรรมเนียมคือว่า เราไม่เขียนเอกลักษณ์

Thai: 
2 คูณ 2 เท่ากับ 1, 0, 0, 1
ธรรมเนียมคือเขียน I2
เพราะเรารู้ว่ามันจะเป็น 2 คูณ 2
มันจะเป็นเมทริกซ์ 2 คูณ 2
มันจะเท่ากับ 1, 0, 0, 1
เอกลักษณ์ 5 จะเป็นเมทริกซ์ 5 คูณ 5
มันจะเป็น 1, 2, 3, 4
0, 1, 2, 1, 3
0, 0, 1, 0, 0
0 -- คุณคงเข้าใจ
0, 0, 0, 1, 0
0, 0, 0, 0, 1
อย่างนั้น
ประเด็นตรงนี้คือสังเกต
ว่าเมทริกซ์จัตุรัส
จะเป็นเมทริกซ์จัตุรัสเสมอ
และมันใช้ได้ถึงแม้คุณจะคูณกับ
เมทริกซ์ที่ไม่ใช่จัตุรัส

Czech: 
je dána prvky
1, 0, 0, 1.
Zásadou zkrátka je, že píšeme jen I2,
protože je pak zřejmé, že bude typu 2x2.
Bude to matice typu 2x2.
Bude mít prvky 1, 0, 0, 1.
I5 pak bude maticí typu 5x5.
Bude dána 1…
Jedna, dva tři, čtyři.
0, 1…dva, jedna, dva, tři.
0, 0, 1, 0, 0,
0…Chápete, co tím myslím…
0, 0, 0, 1, 0,
0, 0, 0, 0, 1.
Asi takhle.
Jde zde jen o to si uvědomit,
že Vaše jednotková matice
bude vždy čtvercovou maticí,
a to i při násobení obdélníkových matic.

English: 
two by two is equal to
one, zero, zero, one.
The convention is actually just write I2
because you know it's
going to be a two by two.
It's going to be a two by two matrix,
it's going to be one, zero, zero, one.
Identity five is going to
be a five by five matrix.
It's going to be one,
one, two, three, four.
Zero, one, two, one, three.
Zero, zero, one, zero, zero.
Zero ... you get the idea,
zero, zero, zero, one, zero.
Zero, zero, zero, zero, one.
Just like that.
The whole point here is just to realize
that your identity matrix
is always going to be a square matrix
and it works even when you're multiplying
non square other matrices.

Bulgarian: 
2 х 2 е равна на 1, 0, 0, 1.
Подходът всъщност е 
просто да пишем I2,
защото знаем, че
ще бъде 2 х 2.
Ще бъде матрица 2 х 2.
Ще бъде 1, 0, 0, 1.
I5 ще бъде матрица 5 х 5.
Тя ще бъде 1, 0, 0, 0, 0,
0, 1, 0, 0, 0,
0, 0, 1, 0, 0,
0...схващаш замисъла.
0, 0, 0, 1, 0,
0, 0, 0, 0, 1.
Ето точно така.
Целият смисъл е 
просто да разберем,
че тази единична матрица
винаги ще бъде
квадратна матрица
и работи дори 
когато умножаваме
неквадратни матрици.

Korean: 
2×2이고, 값은 1, 0, 0, 1이라고 쓰는 것이 아니라
그저 I2라고 작성하는 것입니다
왜냐하면 여러분도 아는듯이 2×2 행렬이
될 것이고,
값은 1, 0, 0, 1이 될 것이기 때문입니다
I5는 5×5 행렬이 될 것입니다
이 행렬은 첫번째 행은 1 뒤로 0이 4개
두 번째는 0, 1, 그 뒤로 0이 3개
그리고, 0, 0, 1, 0, 0
0... 규칙을 이해하셨을 겁니다
0, 0, 0, 1, 0
0, 0, 0, 0, 1
이렇게 될 것입니다
그래서 이 강의의 중점은
그저 단위 행렬이
언제나 정방 행렬이 되고
여러분이 다른 임의의 행렬을
계산해도 그렇다는 것입니다
