that in just about anything you do in life,

在你生活中做的任何事情中。

you need numbers.

你需要數字。

In particular, though,

特別是，雖然。

some fields don't just need a few numbers,

有些資料欄不只需要幾個數字。

they need lots of them.

他們需要很多的人。

How do you keep track of all those numbers?

你是如何記錄所有這些數字的？

Well, mathematicians dating back

數學家們可以追溯到

as early as ancient China

早在中國古代

came up with a way to represent

想出了一個方法來代表

arrays of many numbers at once.

同時有許多數字的數組。

Nowadays we call such an array a "matrix,"

現在我們把這樣的數組稱為"矩陣，&quot。

and many of them hanging out together, "matrices".

和很多人一起出去玩，"矩陣"。

Matrices are everywhere.

矩陣無處不在。

They are all around us,

它們就在我們身邊。

even now in this very room.

即使現在在這個房間裡，

Sorry, let's get back on track.

對不起，讓我們回到正軌。

Matrices really are everywhere, though.

不過，矩陣真的無處不在。

They are used in business,

它們被用於商業。

economics,

經濟學；

cryptography,

密碼學。

physics,

物理學；

electronics,

電子產品。

and computer graphics.

和計算機圖形學。

One reason matrices are so cool

矩陣如此酷的原因之一

is that we can pack so much information into them

是我們可以把這麼多的資訊裝進他們的體內

and then turn a huge series of different problems

然後把一大堆不同的問題

into one single problem.

變成一個單一的問題。

So, to use matrices, we need to learn how they work.

所以，要使用矩陣，我們需要學習矩陣的工作原理。

It turns out, you can treat matrices

原來，你可以把矩陣處理成

just like regular numbers.

就像普通數字一樣。

You can add them,

你可以添加他們。

subtract them,

減去它們。

even multiply them.

甚至乘以它們。

You can't divide them,

你不能把它們分開。

but that's a rabbit hole of its own.

但這'是自己的兔子洞。

Adding matrices is pretty simple.

添加矩陣非常簡單。

All you have to do is add the corresponding entries

您所要做的就是添加相應的條目。

in the order they come.

按照他們來的順序。

So the first entries get added together,

所以第一條就被加在一起。

the second entries,

第二條。

the third,

第三個。

all the way down.

一路下來。

Of course, your matrices have to be the same size,

當然，你的矩陣必須是相同的大小。

but that's pretty intuitive anyway.

但無論如何，這'很直觀。

You can also multiply the whole matrix

您也可以將整個矩陣乘以

by a number, called a scalar.

由一個數字，稱為標量。

Just multiply every entry by that number.

只需將每個條目乘以這個數字。

But wait, there's more!

但是，等一下，還有更多!

You can actually multiply one matrix by another matrix.

實際上你可以用一個矩陣乘以另一個矩陣。

It's not like adding them, though,

不過，它'並不像添加它們。

where you do it entry by entry.

你在哪裡逐條做。

It's more unique

它更獨特

and pretty cool once you get the hang of it.

一旦你掌握了訣竅，就會變得非常酷。

Here's how it works.

這裡'是如何工作的。

Let's say you have two matrices.

讓我們'說你有兩個矩陣。

Let's make them both two by two,

讓我們'讓他們兩兩相爭。

meaning two rows by two columns.

意思是兩行兩列。

Write the first matrix to the left

將第一個矩陣寫在左邊

and the second matrix goes next to it

而第二個矩陣就在它旁邊

and translated up a bit,

並翻譯了一下。

kind of like we are making a table.

有點像我們在做一張桌子。

The product we get when we multiply the matrices together

我們將矩陣相乘後得到的乘積。

will go right between them.

會在他們之間進行。

We'll also draw some gridlines to help us along.

我們'也會畫一些網格線來幫助我們。

Now, look at the first row of the first matrix

現在，看看第一個矩陣的第一行。

and the first column of the second matrix.

和第二矩陣的第一列。

See how there's two numbers in each?

看到每個人都有兩個數字了嗎？

Multiply the first number in the row

乘以行中的第一個數字

by the first number in the column:

由列中的第一個數字。

1 times 2 is 2.

1乘以2就是2。

Now do the next ones:

現在做下一個。

3 times 3 is 9.

3乘以3是9。

Now add them up:

現在把它們加起來。

2 plus 9 is 11.

2加9是11。

Let's put that number in the top-left position

讓我們把這個數字放在左上角的位置。

so that it matches up with the rows and columns

以使其與行和列相匹配

we used to get it.

我們曾經得到它。

See how that works?

看到了嗎？

You can do the same thing to get the other entries.

你可以做同樣的事情來獲得其他條目。

-4 plus 0 is -4.

-4加0就是-4。

4 plus -3 is 1.

4加-3是1。

-8 plus 0 is -8.

-8加0就是-8。

So, here's your answer.

所以，這就是你的答案。

Not all that bad, is it?

也不至於那麼糟糕吧？

There's one catch, though.

不過有一個問題。

Just like with addition,

就像用加法一樣。

your matrices have to be the right size.

你的矩陣必須是正確的大小。

Look at these two matrices.

看看這兩個矩陣。

2 times 8 is 16.

2乘以8是16。

3 times 4 is 12.

3乘以4是12。

3 times

3次

wait a minute,

等一下

there are no more rows in the second matrix.

在第二個矩陣中沒有更多的行。

We ran out of room.

我們沒有房間了。

So, these matrices can't be multiplied.

所以，這些矩陣是不能相乘的'。

The number of columns in the first matrix

第一個矩陣中的列數

has to be the same as the number of rows in the second matrix.

必須與第二個矩陣的行數相同。

As long as you're careful

只要你小心

to match up your dimensions right, though,

以配合你的尺寸，雖然。

it's pretty easy.

這很容易。

Understanding matrix multiplication

瞭解矩陣乘法

is just the beginning, by the way.

對了，這只是個開始。

There's so much you can do with them.

有'的這麼多，你可以做他們。

For example, let's say you want

例如，讓我們說你想

to encrypt a secret message.

加密密文。

Let's say it's "Math rules".

比方說是'的"數學規則"。

Though, why anybody would want to keep this a secret

不過，為什麼有人要保守這個祕密呢？

is beyond me.

是超越我。

Letting numbers stand for letters,

讓數字代表字母。

you can put the numbers in a matrix

你可以把這些數字放在一個矩陣中

and then an encryption key in another.

然後在另一個加密密鑰。

Multiply them together

將它們相乘

and you've got a new encoded matrix.

你就得到了一個新的編碼矩陣。

The only way to decode the new matrix

解碼新矩陣的唯一方法

and read the message

並閱讀資訊

is to have the key,

是擁有鑰匙。

that second matrix.

這第二個矩陣。

There's even a branch of mathematics

甚至還有一個數學分支。

that uses matrices constantly,

不斷使用矩陣的。

called Linear Algebra.

稱為線性代數。

If you ever get a chance to study Linear Algebra,

如果你有機會學習線性代數。

do it, it's pretty awesome.

做到這一點，它'是相當不錯的。

But just remember,

但只要記住。

once you know how to use matrices,

一旦你知道如何使用矩陣。

you can do pretty much anything.

你可以做幾乎任何事情。