非方阵作为维度间的变换 | 第8章，线性代数的本质：英语精读讲义

Hey everyone,

大家好，

I've got another quick footnote for you between chapters today. 表達脈絡

今天在两章之间，我给大家准备了一个简短的补充说明。

When I've talked about linear transformations so far,

到目前为止，当我谈论线性变换时，

I've only really talked about transformations from 2D vectors to other 2D vectors

我只讨论了从2D向量到其他2D向量的变换，

represented with 2x2 matrices

即用2x2矩阵表示的变换，

or from 3D vectors to other 3D vectors

或者从3D向量到其他3D向量的变换，

represented with 3x3 matrices.

即用3x3矩阵表示的变换。

But several commenters have asked about non-square matrices, 文化背景

但是一些评论者问到了非方阵的问题，

so I thought I'd take a moment to just show what those mean geometrically. 講解

所以我想花点时间从几何角度展示它们的含义。

By now in the series,

到了本系列的这个阶段，

you actually have most of the background you need to start pondering a question like this on your own, 短語搭配

大家其实已经具备了大部分必要的背景知识，可以开始自己思考这样的问题了，

but I'll start talking through it just to give a little mental momentum. 講解

但我还是先讲解一下，给大家提供一点思路上的推动力。

It's perfectly reasonable to talk about transformations between dimensions 短語搭配

谈论不同维度之间的变换是完全合理的，

such as one that takes 2D vectors to 3D vectors.

比如将2D向量转换为3D向量的变换。

Again, what makes one of these linear is that gridlines remain parallel and evenly spaced 語法

同样，使这些变换保持线性的条件是网格线保持平行且等距，

and that the origin maps to the origin. 用詞

并且原点映射到原点。

What I have pictured here is the input space on the left, 講解

我在这里展示的是左侧的输入空间，

which is just 2D space,

它只是一个2D空间，

and the output of the transformation shown on the right.

而变换的输出显示在右侧。

The reason I'm not showing the inputs move over to the outputs like I usually do 語法

我没有像往常那样展示输入移动到输出的过程，

is not just animation laziness.

这不仅仅是因为动画制作上想偷懒。

It's worth emphasizing that 2D vector inputs are very different animals from these 3D vector outputs 講解

需要强调的是，2D向量输入与这些3D向量输出是完全不同的生物，

living in a completely separate unconnected space.

它们存在于完全独立且不相连的空间中。

Encoding one of these transformations with a matrix is really just the same thing as what we've done before. 短語搭配

用矩阵来编码这些变换，实际上和我们以前做的一样。

You look at where each basis vector lands 講解

你观察每个基向量落脚的位置，

and write the coordinates of the landing spots as the columns of a matrix. 短語搭配

并将落脚点的坐标写成矩阵的列。

For example, what you're looking at here is an output of a transformation that takes i-hat

例如，你在这里看到的是一个变换的输出，它将i帽

to the coordinates 2 -1 -2

映射到坐标 2 -1 -2，

and j-hat to the coordinates 0 1 1.

将j帽映射到坐标 0 1 1。

Notice, this means the matrix encoding our transformation has three rows and two columns, 表達脈絡

注意，这意味着编码我们变换的矩阵有三行两列，

which, to use standard terminology, makes it a 3x2 matrix. 講解

用标准术语来说，这就是一个3x2矩阵。

In the language of last video, 表達脈絡

用上一个视频的术语来说，

the column space of this matrix, the place where all the vectors land, 文化背景

这个矩阵的列空间，也就是所有向量落脚的地方，

is a 2D plane slicing through the origin of 3D space. 用詞

是一个穿过3D空间原点的2D平面。

But the matrix is still full rank since the number of dimensions in this column space 文化背景

但这个矩阵仍然是满秩的，因为这个列空间的维数

is the same as the number of dimensions of the input space.

与输入空间的维数相同。

So, if you see a 3x2 matrix out in the wild, 講解

所以，如果你在实际中看到一个3x2矩阵，

you can know that it has the geometric interpretation of mapping two dimensions to three dimensions.

你就能知道它的几何意义是将二维空间映射到三维空间。

Since the two columns indicate that the input space has two basis vectors

因为这两列表明输入空间有两个基向量，

and the three rows indicate that the landing spots for each of those basis vectors

而这三行表明每个基向量的落脚点

is described with three separate coordinates.

都是用三个独立的坐标来描述的。

Likewise, if you see a 2x3 matrix with two rows and three columns,

同样地，如果你看到一个有两行三列的2x3矩阵，

what do you think that means?

你认为那意味着什么？

Well, the three columns indicate that you're starting in a space that has three basis vectors,

嗯，这三列表明你从一个有三个基向量的空间开始，

so we're starting in three dimensions.

所以我们是从三维空间开始的。

And the two rows indicate that the landing spot for each of those three basis vectors

而这两行表明这三个基向量中每一个的落脚点

is described with only two coordinates,

都仅用两个坐标来描述，

so they must be landing in two dimensions. 語法

所以它们一定落在二维空间中。

So, it's a transformation from 3D space onto the 2D plane. 文化背景

因此，这是一个从3D空间到2D平面的变换。

A transformation that should feel very uncomfortable if you imagine going through it. 習語

如果你想象自己经历这个变换，应该会觉得非常不舒服。

You could also have a transformation from two dimensions to one dimension.

你也可以有一个从二维到一维的变换。

One-dimensional space is really just the number line, 文化背景

一维空间其实就是数轴，

so a transformation like this takes in 2D vectors and spits out numbers. 短語動詞

所以这样的变换接收2D向量，然后吐出数字。

Thinking about grid lines remaining parallel and evenly spaced is a little bit messy due to all of the squishification happening here. 用詞

考虑到这里发生的各种压缩，想象网格线保持平行且等距会稍微有些混乱。

So, in this case,

因此，在这种情况下，

the visual understanding for what linearity means is that if you have a line of evenly spaced dots, 語法

线性含义的直观理解是，如果你有一排等距的点，

it would remain evenly spaced once they're mapped onto the number line.

一旦它们被映射到数轴上，它们将保持等距。

One of these transformations is encoded with a 1x2 matrix,

其中一个变换可以用一个1x2矩阵编码，

each of whose two columns has just a single entry. 講解

它的两列中每列只有一个元素。

The two columns represent where the basis vectors land,

这两列代表了基向量的落脚点，

and each one of those columns requires just one number,

而每一列只需要一个数字，

the number that that basis vector landed on.

即那个基向量所落在的数字。

This is actually a surprisingly meaningful type of transformation with close ties to the dot product, 短語搭配

这实际上是一种出人意料地有意义的变换，与点积有着密切的联系，

and I'll be talking about that next video.

我将在下个视频中讨论这个问题。

Until then, I encourage you to play around with this idea on your own, 講解

在此之前，我鼓励你们自己尝试一下这个想法，

contemplating the meanings of things like matrix multiplication and linear systems of equations

在不同维度之间变换的背景下，

in the context of transformations between different dimensions. 短語搭配

思考矩阵乘法和线性方程组等概念的意义。

Have fun.

玩得开心。