三維線性變換 | 第5章，線性代數的本質：英語精讀講義

Hey folks, I've got a relatively quick video for you today,

大家好，今天為大家帶來一個相對簡短的影片，

just sort of a footnote between chapters. 講解

就像是章節之間的一個小註腳。

In the last two videos, I talked about linear transformations and matrices,

在前兩支影片中，我談到了線性變換和矩陣，

but I only showed the specific case of transformations

但我只展示了特定情況下的變換，

that take two-dimensional vectors to other two-dimensional vectors.

即將二維向量映射為其他二維向量。

In general throughout this series, we'll work mainly in two dimensions,

通常在這個系列中，我們主要在二維空間中進行工作，

mostly because it's easier to actually see on the screen and wrap your mind around. 慣用語

主要是因為它在螢幕上更容易被看到，也更容易理解。

But more importantly than that, 表達脈絡

但更重要的是，

once you get all the core ideas in two dimensions,

一旦你掌握了二維空間中的所有核心概念，

they carry over pretty seamlessly to higher dimensions. 片語動詞

它們可以非常無縫地延伸到更高維度。

Nevertheless, it's good to peek our heads outside of flatland now and then 講解

儘管如此，我們偶爾也應該把目光投向「平面國」之外，

to, you know, see what it means to apply these ideas

去看看應用這些想法意味著什麼

in more than just those two dimensions. 文法

在超過二維的空間中。

For example, consider a linear transformation

比如，考慮一個線性變換，

with three-dimensional vectors as inputs

其輸入是三維向量，

and three-dimensional vectors as outputs.

輸出也是三維向量。

We can visualize this by smooshing around all the points in three-dimensional space 用詞

我們可以透過擠壓三維空間中的所有點來視覺化這一點，

as represented by a grid in such a way that keeps the grid lines parallel and evenly spaced 文法

該空間由網格表示，變換需要保持網格線平行且等距，

and which fixes the origin in place. 用詞

並且保持原點固定不動。

And just as with two dimensions,

就像二維空間一樣，

every point of space that we see moving around is really just a proxy for a vector 用詞

我們看到在空間中移動的每一個點，其實都代表著一個向量，

who has its tip at that point

它的終點就在那個點上，

and what we're really doing is thinking about input vectors moving over to their corresponding outputs. 表達脈絡

我們真正在思考的是，輸入向量移動到了它們對應的輸出向量。

And just as with two dimensions,

同樣，正如在二維空間中一樣，

one of these transformations is completely described by where the basis vectors go. 片語搭配

這些變換之一可以完全由基底向量的去向來描述。

But now, there are three standard basis vectors that we typically use.

但現在，我們通常使用三個標準基底向量。

The unit vector in the x direction, i-hat,

x 方向的單位向量，i-hat，

the unit vector in the y direction, j-hat,

y 方向的單位向量，j-hat，

and a new guy, 用詞

以及一個新夥伴，

the unit vector in the z direction, called k-hat.

z 方向的單位向量，稱為 k-hat。

In fact, I think it's easier to think about these transformations by only following those basis vectors

實際上，我覺得透過只追蹤這些基底向量來思考這些變換會更容易，

since the full 3D grid representing all points can get kind of messy.

因為代表所有點的完整 3D 網格可能會變得有些凌亂。

By leaving a copy of the original axes in the background, 文法

透過在背景中保留一份原始座標軸的副本，

we can think about the coordinates of where each of these three basis vectors lands. 用詞

我們就能思考這三個基底向量各自落在哪個座標上。

Record the coordinates of these three vectors as the columns of a 3x3 matrix.

將這三個向量的座標記錄為一個 3x3 矩陣的行（column）。

This gives a matrix that completely describes the transformation using only nine numbers.

這就得到了一個僅用9個數字就完全描述了該變換的矩陣。

As a simple example,

舉個簡單的例子，

consider the transformation that rotates space 90 degrees around the y-axis.

考慮將空間繞 y 軸旋轉 90 度的變換。

So that would mean that it takes i-hat to the coordinates 0 0 -1 on the z-axis. 用詞

這意味著它將 i-hat 變到 z 軸上的座標 0 0 -1。

It doesn't move j-hat,

它不移動 j-hat，

so it stays at the coordinates 0 1 0,

所以它留在座標 0 1 0，

and then k-hat moves over to the x-axis at 1 0 0.

然後 k-hat 移動到 x 軸上的 1 0 0。

Those three sets of coordinates become the columns of a matrix that describes that rotation transformation.

這三組座標就成了描述該旋轉變換矩陣的行。

To see where a vector with coordinates x y z lands,

要看一個座標為 x y z 的向量落在哪，

the reasoning is almost identical to what it was for two dimensions. 片語搭配

推理過程和二維時幾乎一模一樣。

Each of those coordinates can be thought of as instructions for how to scale each basis vector 講解

這些座標中的每一個都可以看作是對每個基底向量進行縮放的指令，

so that they add together to get your vector.

從而讓它們相加得到你的向量。

And the important part, just like the 2D case, 表達脈絡

就像 2D 情況一樣，最重要的是，

is that this scaling and adding process works both before and after the transformation. 文法

這個縮放和相加的過程在變換前後都是成立的。

So to see where your vector lands,

所以要看你的向量落在哪裡，

you multiply those coordinates by the corresponding columns of the matrix 片語搭配

你只需將這些座標乘以矩陣對應的行，

and then you add together the three results.

然後把這三個結果加起來。

Multiplying two matrices is also similar.

兩個矩陣相乘的情況也類似。

Whenever you see two 3x3 matrices getting multiplied together, 文法

每當你看到兩個 3x3 矩陣相乘時，

you should imagine first applying the transformation encoded by the right one, 用詞

你應該想像先應用右邊那個矩陣編碼的變換，

then applying the transformation encoded by the left one.

然後再應用左邊那個矩陣編碼的變換。

It turns out that 3D matrix multiplication is actually pretty important for fields like computer graphics and robotics 表達脈絡

事實證明，3D 矩陣乘法對於電腦圖形學和機器人技術等領域其實非常重要，

since things like rotations in three dimensions can be pretty hard to describe,

因為像三維空間中的旋轉這樣的東西描述起來可能相當困難，

but they're easier to wrap your mind around if you can break them down 片語動詞

但如果能將它們分解，理解起來就容易多了

as the composition of separate, easier to think about rotations. 用詞

分解為獨立的、更容易思考的旋轉的複合。

Performing this matrix multiplication numerically is, once again,

同樣地，在數值上執行這種矩陣乘法

pretty similar to the two-dimensional case.

與二維空間的情況非常相似。

In fact, a good way to test your understanding of the last video 文法

實際上，檢驗你對上支影片理解程度的一個好方法

would be to try to reason through what specifically this matrix multiplication should look like, 片語動詞

是試著推導這種矩陣乘法具體應該是什麼樣子的，

thinking closely about how it relates to the idea of applying two successive transformations in space.

仔細思考它與在空間中連續應用兩次變換的概念有何聯繫。

In the next video,

在下一支影片中，

I'll start getting into the determinant. 片語動詞

我將開始介紹行列式。