非方陣作為維度間的變換 | 第8章，線性代數的本質：英語精讀講義

Hey everyone,

大家好，

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

今天在兩章之間，我給大家準備了一個簡短的補充說明。

When I've talked about linear transformations so far,

到目前為止，當我談論線性變換時，

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

我只討論了從2D向量到其他2D向量的變換，

represented with 2x2 matrices

即用2x2矩陣表示的變換，

or from 3D vectors to other 3D vectors

或者從3D向量到其他3D向量的變換，

represented with 3x3 matrices.

即用3x3矩陣表示的變換。

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

但是一些評論者問到了非方陣的問題，

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

所以我想花點時間從幾何角度展示它們的含義。

By now in the series,

到了本系列的這個階段，

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

大家其實已經具備了大部分必要的背景知識，可以開始自己思考這樣的問題了，

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

但我還是先講解一下，給大家提供一點思路上的推動力。

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

談論不同維度之間的變換是完全合理的，

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

比如將2D向量轉換為3D向量的變換。

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

同樣，使這些變換保持線性的條件是網格線保持平行且等距，

and that the origin maps to the origin. 用詞

並且原點映射到原點。

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

我在這裡展示的是左側的輸入空間，

which is just 2D space,

它只是一個2D空間，

and the output of the transformation shown on the right.

而變換的輸出顯示在右側。

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

我沒有像往常那樣展示輸入移動到輸出的過程，

is not just animation laziness.

這不僅僅是因為動畫製作上想偷懶。

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

需要強調的是，2D向量輸入與這些3D向量輸出是完全不同的生物，

living in a completely separate unconnected space.

它們存在於完全獨立且不相連的空間中。

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

用矩陣來編碼這些變換，實際上和我們以前做的一樣。

You look at where each basis vector lands 講解

你觀察每個基向量落腳的位置，

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

並將落腳點的座標寫成矩陣的行。

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

例如，你在這裡看到的是一個變換的輸出，它將i帽

to the coordinates 2 -1 -2

映射到座標 2 -1 -2，

and j-hat to the coordinates 0 1 1.

將j帽映射到座標 0 1 1。

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

注意，這意味著編碼我們變換的矩陣有三列兩行，

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

用標準術語來說，這就是一個3x2矩陣。

In the language of last video, 表達脈絡

用上一個影片的術語來說，

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

這個矩陣的行空間，也就是所有向量落腳的地方，

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

是一個穿過3D空間原點的2D平面。

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

但這個矩陣仍然是滿秩的，因為這個行空間的維數

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

與輸入空間的維數相同。

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

所以，如果你在實際中看到一個3x2矩陣，

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

你就能知道它的幾何意義是將二維空間映射到三維空間。

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

因為這兩行表明輸入空間有兩個基向量，

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

而這三列表明每個基向量的落腳點

is described with three separate coordinates.

都是用三個獨立的座標來描述的。

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

同樣地，如果你看到一個有兩列三行的2x3矩陣，

what do you think that means?

你認為那意味著什麼？

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

嗯，這三行表明你從一個有三個基向量的空間開始，

so we're starting in three dimensions.

所以我們是從三維空間開始的。

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

而這兩列表明這三個基向量中每一個的落腳點

is described with only two coordinates,

都僅用兩個座標來描述，

so they must be landing in two dimensions. 文法

所以它們一定落在二維空間中。

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

因此，這是一個從3D空間到2D平面的變換。

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

如果你想像自己經歷這個變換，應該會覺得非常不舒服。

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

你也可以有一個從二維到一維的變換。

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

一維空間其實就是數線上，

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

所以這樣的變換接收2D向量，然後吐出數字。

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

考慮到這裡發生的各種壓縮，想像網格線保持平行且等距會稍微有些混亂。

So, in this case,

因此，在這種情況下，

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

線性含義的直觀理解是，如果你有一排等距的點，

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

一旦它們被映射到數線上，它們將保持等距。

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

其中一個變換可以用一個1x2矩陣編碼，

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

它的兩行中每行只有一個元素。

The two columns represent where the basis vectors land,

這兩行代表了基向量的落腳點，

and each one of those columns requires just one number,

而每一行只需要一個數字，

the number that that basis vector landed on.

即那個基向量所落下的數字。

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

這實際上是一種出人意料地有意義的變換，與內積有著密切的聯繫，

and I'll be talking about that next video.

我將在下個影片中討論這個問題。

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

在此之前，我鼓勵你們自己嘗試一下這個想法，

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

在不同維度之間變換的背景下，

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

思考矩陣乘法和線性方程組等概念的意義。

Have fun.

玩得開心。