三维线性变换 | 第5章，线性代数的本质：英语精读讲义

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

大家好，今天给大家带来一个相对简短的视频，

just sort of a footnote between chapters. 講解

就像是章节之间的一个小注脚。

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

在前两个视频中，我谈到了线性变换和矩阵，

but I only showed the specific case of transformations

但我只展示了特定情况下的变换，

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

即将二维向量映射为其他二维向量。

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

通常在这个系列中，我们主要在二维空间中进行工作，

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

主要是因为它在屏幕上更容易被看到，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,

一旦你掌握了二维空间中的所有核心概念，

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

它们可以非常无缝地延伸到更高维度。

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

尽管如此，我们偶尔也应该把目光投向“平面国”之外，

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

去看看应用这些想法意味着什么

in more than just those two dimensions. 語法

在超过二维的空间中。

For example, consider a linear transformation

比如，考虑一个线性变换，

with three-dimensional vectors as inputs

其输入是三维向量，

and three-dimensional vectors as outputs.

输出也是三维向量。

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

我们可以通过挤压三维空间中的所有点来可视化这一点，

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

该空间由网格表示，变换需要保持网格线平行且等距，

and which fixes the origin in place. 用詞

并且保持原点固定不动。

And just as with two dimensions,

就像二维空间一样，

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

我们看到在空间中移动的每一个点，其实都代表着一个向量，

who has its tip at that point

它的终点就在那个点上，

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

我们真正在思考的是，输入向量移动到了它们对应的输出向量。

And just as with two dimensions,

同样，正如在二维空间中一样，

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

这些变换之一可以完全由基向量的去向来描述。

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

但现在，我们通常使用三个标准基向量。

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

x 方向的单位向量，i-hat，

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

y 方向的单位向量，j-hat，

and a new guy, 用詞

以及一个新伙伴，

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

z 方向的单位向量，称为 k-hat。

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

实际上，我觉得通过只追踪这些基向量来思考这些变换会更容易，

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

因为代表所有点的完整 3D 网格可能会变得有些凌乱。

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

通过在背景中保留一份原始坐标轴的副本，

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

我们就能思考这三个基向量各自落在哪个坐标上。

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

将这三个向量的坐标记录为一个 3x3 矩阵的列。

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

这就得到了一个仅用9个数字就完全描述了该变换的矩阵。

As a simple example,

举个简单的例子，

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

考虑将空间绕 y 轴旋转 90 度的变换。

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

这意味着它将 i-hat 变到 z 轴上的坐标 0 0 -1。

It doesn't move j-hat,

它不移动 j-hat，

so it stays at the coordinates 0 1 0,

所以它留在坐标 0 1 0，

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

然后 k-hat 移动到 x 轴上的 1 0 0。

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

这三组坐标就成了描述该旋转变换矩阵的列。

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

要看一个坐标为 x y z 的向量落在哪，

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

推理过程和二维时几乎一模一样。

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

这些坐标中的每一个都可以看作是对每个基向量进行缩放的指令，

so that they add together to get your vector.

从而让它们相加得到你的向量。

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

就像 2D 情况一样，最重要的是，

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

这个缩放和相加的过程在变换前后都是成立的。

So to see where your vector lands,

所以要看你的向量落在哪里，

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

你只需将这些坐标乘以矩阵对应的列，

and then you add together the three results.

然后把这三个结果加起来。

Multiplying two matrices is also similar.

两个矩阵相乘的情况也类似。

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

每当你看到两个 3x3 矩阵相乘时，

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

你应该想象先应用右边那个矩阵编码的变换，

then applying the transformation encoded by the left one.

然后再应用左边那个矩阵编码的变换。

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

事实证明，3D 矩阵乘法对于计算机图形学和机器人技术等领域其实非常重要，

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

因为像三维空间中的旋转这样的东西描述起来可能相当困难，

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

但如果能将它们分解，理解起来就容易多了

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

分解为独立的、更容易思考的旋转的复合。

Performing this matrix multiplication numerically is, once again,

同样地，在数值上执行这种矩阵乘法

pretty similar to the two-dimensional case.

与二维空间的情况非常相似。

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

实际上，检验你对上个视频理解程度的一个好方法

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

是试着推导这种矩阵乘法具体应该是什么样子的，

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

仔细思考它与在空间中连续应用两次变换的概念有何联系。

In the next video,

在下一个视频中，

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

我将开始介绍行列式。