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By now, I'm sure you know
現在，我相信你已經知道了。
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that in just about anything you do in life,
在你生活中做的任何事情中。
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you need numbers.
你需要數字。
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In particular, though,
特別是，雖然。
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some fields don't just need a few numbers,
有些資料欄不只需要幾個數字。
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they need lots of them.
他們需要很多的人。
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How do you keep track of all those numbers?
你是如何記錄所有這些數字的？
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Well, mathematicians dating back
數學家們可以追溯到
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as early as ancient China
早在中國古代
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came up with a way to represent
想出了一個方法來代表
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arrays of many numbers at once.
同時有許多數字的數組。
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Nowadays we call such an array a "matrix,"
現在我們把這樣的數組稱為"矩陣，"。
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and many of them hanging out together, "matrices".
和很多人一起出去玩，"矩陣"。
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Matrices are everywhere.
矩陣無處不在。
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They are all around us,
它們就在我們身邊。
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even now in this very room.
即使現在在這個房間裡，
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Sorry, let's get back on track.
對不起，讓我們回到正軌。
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Matrices really are everywhere, though.
不過，矩陣真的無處不在。
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They are used in business,
它們被用於商業。
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economics,
經濟學；
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cryptography,
密碼學。
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physics,
物理學；
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electronics,
電子產品。
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and computer graphics.
和計算機圖形學。
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One reason matrices are so cool
矩陣如此酷的原因之一
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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
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and then turn a huge series of different problems
然後把一大堆不同的問題
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into one single problem.
變成一個單一的問題。
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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。
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It turns out, you can treat matrices
原來，你可以把矩陣處理成
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just like regular numbers.
就像普通數字一樣。
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You can add them,
你可以添加他們。
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subtract them,
減去它們。
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even multiply them.
甚至乘以它們。
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You can't divide them,
你不能把它們分開。
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but that's a rabbit hole of its own.
但這'是自己的兔子洞。
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Adding matrices is pretty simple.
添加矩陣非常簡單。
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All you have to do is add the corresponding entries
您所要做的就是添加相應的條目。
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in the order they come.
按照他們來的順序。
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So the first entries get added together,
所以第一條就被加在一起。
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the second entries,
第二條。
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the third,
第三個。
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all the way down.
一路下來。
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Of course, your matrices have to be the same size,
當然，你的矩陣必須是相同的大小。
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but that's pretty intuitive anyway.
但無論如何，這'很直觀。
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You can also multiply the whole matrix
您也可以將整個矩陣乘以
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by a number, called a scalar.
由一個數字，稱為標量。
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Just multiply every entry by that number.
只需將每個條目乘以這個數字。
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But wait, there's more!
但是，等一下，還有更多!
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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。
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It's not like adding them, though,
不過，它'並不像添加它們。
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where you do it entry by entry.
你在哪裡逐條做。
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It's more unique
它更獨特
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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。
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Here's how it works.
這裡'是如何工作的。
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Let's say you have two matrices.
讓我們'說你有兩個矩陣。
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Let's make them both two by two,
讓我們'讓他們兩兩相爭。
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meaning two rows by two columns.
意思是兩行兩列。
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Write the first matrix to the left
將第一個矩陣寫在左邊
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and the second matrix goes next to it
而第二個矩陣就在它旁邊
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and translated up a bit,
並翻譯了一下。
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kind of like we are making a table.
有點像我們在做一張桌子。
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The product we get when we multiply the matrices together
我們將矩陣相乘後得到的乘積。
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will go right between them.
會在他們之間進行。
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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。
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Now, look at the first row of the first matrix
現在，看看第一個矩陣的第一行。
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and the first column of the second matrix.
和第二矩陣的第一列。
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See how there's two numbers in each?
看到每個人都有兩個數字了嗎？
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Multiply the first number in the row
乘以行中的第一個數字
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by the first number in the column:
由列中的第一個數字。
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1 times 2 is 2.
1乘以2就是2。
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Now do the next ones:
現在做下一個。
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3 times 3 is 9.
3乘以3是9。
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Now add them up:
現在把它們加起來。
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2 plus 9 is 11.
2加9是11。
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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。
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so that it matches up with the rows and columns
以使其與行和列相匹配
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we used to get it.
我們曾經得到它。
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See how that works?
看到了嗎？
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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。
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-4 plus 0 is -4.
-4加0就是-4。
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4 plus -3 is 1.
4加-3是1。
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-8 plus 0 is -8.
-8加0就是-8。
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So, here's your answer.
所以，這就是你的答案。
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Not all that bad, is it?
也不至於那麼糟糕吧？
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There's one catch, though.
不過有一個問題。
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Just like with addition,
就像用加法一樣。
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your matrices have to be the right size.
你的矩陣必須是正確的大小。
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Look at these two matrices.
看看這兩個矩陣。
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2 times 8 is 16.
2乘以8是16。
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3 times 4 is 12.
3乘以4是12。
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3 times
3次
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wait a minute,
等一下
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there are no more rows in the second matrix.
在第二個矩陣中沒有更多的行。
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We ran out of room.
我們沒有房間了。
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So, these matrices can't be multiplied.
所以，這些矩陣是不能相乘的'。
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The number of columns in the first matrix
第一個矩陣中的列數
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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。
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As long as you're careful
只要你小心
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to match up your dimensions right, though,
以配合你的尺寸，雖然。
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it's pretty easy.
這很容易。
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Understanding matrix multiplication
瞭解矩陣乘法
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is just the beginning, by the way.
對了，這只是個開始。
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There's so much you can do with them.
有'的這麼多，你可以做他們。
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For example, let's say you want
例如，讓我們說你想
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to encrypt a secret message.
加密密文。
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Let's say it's "Math rules".
比方說是'的"數學規則"。
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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？
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is beyond me.
是超越我。
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Letting numbers stand for letters,
讓數字代表字母。
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you can put the numbers in a matrix
你可以把這些數字放在一個矩陣中
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and then an encryption key in another.
然後在另一個加密密鑰。
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Multiply them together
將它們相乘
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and you've got a new encoded matrix.
你就得到了一個新的編碼矩陣。
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The only way to decode the new matrix
解碼新矩陣的唯一方法
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and read the message
並閱讀資訊
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is to have the key,
是擁有鑰匙。
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that second matrix.
這第二個矩陣。
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There's even a branch of mathematics
甚至還有一個數學分支。
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that uses matrices constantly,
不斷使用矩陣的。
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called Linear Algebra.
稱為線性代數。
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If you ever get a chance to study Linear Algebra,
如果你有機會學習線性代數。
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do it, it's pretty awesome.
做到這一點，它'是相當不錯的。
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But just remember,
但只要記住。
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once you know how to use matrices,
一旦你知道如何使用矩陣。
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you can do pretty much anything.
你可以做幾乎任何事情。
