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• Hey, everyone!

嗨，大家好！

• I've got another quick footnote for you between chapters today.

今天在章節之閒的我給你們一個很簡單的注解。

• When I talked about linear transformation so far,

至今已來在我講到綫性變換的時候，

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

其實我只講到以2 x 2矩陣所代表著的

• represented with 2-by-2 matrices;

一些2-維矢量變換到另一些2-維矢量；

• or from 3-D vectors to other 3-D vectors, represented with 3-by-3 matrices.

或者以3 x 3矩陣代表的一些3-維矢量變換到其它的矢量。

而有好多人問起非正方的矩陣，

• so I thought I'd take a moment to just show with those means geometrically.

所以我想就花些時間來顯示它們的幾何意義。

• By now in the series, you actually have most of the background you need

在這個系列裏的現在，你們實際上已經有了你們所需要的大多數的背景知識

• to start pondering a question like this on your own.

你們自己開始來考慮像這種問題。

• But I'll start talking through it, just to give a little mental momentum.

而我開始講一些，不過是給一點思考上的動力。

• It's perfectly reasonable to talk about transformations between dimensions,

在不同的維數之間來講講變換是完全合理的，

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

比方說把一些2-維的矢量變到一些3-維的矢量

• Again, what makes one of these linear

再提一下，使變換是綫性的就是

• is that grid lines remain parallel and evenly spaced, and that the origin maps to the origin.

網格保持平行並均等，和原點仍舊不變。

• What I have pictured here is the input space on the left, which is just 2-D space,

我在這裏所畫的，左面是輸入空間，它只是一個2-維的空間，

• and the output of the transformation shown on the right.

而變換的輸出在右面。

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

我沒有，像通常我畫出輸入移動到輸出

• is not just animation laziness.

的道理倒不單是懶得畫動畫。

• It's worth emphasizing the 2-D vector inputs are very different animals from these 3-D

這值得强調輸入的2-維矢量和輸出的

• vector outputs,

3維矢量可是完全不同的東西，

• living in a completely separate unconnected space.

它們存在於一個完全分開的，不相連的空間。

• Encoding one of these transformations with a matrix is really just the same thing as

以矩陣來記錄這些變換中的一個正是和

• what we've done before.

你們以前已經做個的一樣的事。

• You look at where each basis vector lands

你看著各個單位矢量所停留的地方

• and write the coordinates of the landing spots as the columns of a matrix.

並把它停著的地方的坐標寫作一個矩陣的那些列。

• For example, what you're looking at here is an output of a transformation

舉個例子，在這裏你們現在看到的是一個變換的輸出

• that takes i-hat to the coordinates (2, -1, -2) and j-hat to the coordinates (0, 1, 1).

它把i-hat放到坐標(2, -1, -2)和j-hat放到(0, 1, 1).

• Notice, this means the matrix encoding our transformation has 3 rows and 2 columns,

注意，這個記錄著我們變換的矩陣有3個行和2個列，

• which, to use standard terminology, makes it a 3-by-2 matrix.

這用標準的術語，它就是一個 3x2的矩陣。

• In the language of last video, the column space of this matrix,

以上一個錄像中所用的語言，這個矩陣的列空間，

• the place where all the vectors land is a 2-D plane slicing through the origin of 3-D

這個説要矢量所停下的地方是通過3-維空間的原點一個2-維平面

• space.

但是這個矩陣仍是一個全秩（full rank）的，

• But the matrix is still full rank,

因爲在這個列空間中的維數和輸入

• since the number of dimensions in this column space is the same as the number of dimensions

空間的維數是相同的。

• of the input space.

所以，如果你看到有一個3x2的矩陣的存在，

• So, if you see a 3-by-2 matrix out in the wild,

你能知道它有映射(mapping)2-維到3-維的幾何解釋。

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

因爲2個列指出輸入空間有2個單位矢量，

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

而2個行指出各個單位矢量所停下的那些點上

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

是由3個分開的坐標來規定的。

• is described with three separate coordinates.

於此類似的，如果你們看到一個2x3的矩陣，2個行和3個列，你考慮一下

• Likewise, if you see a 2-by-3 matrix with two rows and three columns, what do you think

它的意思是什麽呢？

• that means?

好吧，這3個列指出你們在一個有3個單位矢量的空間中開始，

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

所以我們在3-維中開始，

• so we're starting in three dimensions;

而2個行指出這3個單位矢量停下的點

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

卻只有用2個坐標來描述的，

• is described with only two coordinates,

所以它們一定是停在2-維的了。

• so they must be landing in two dimensions.

因此這是從3-維空間到2-維平面的一個變換。

• So it's a transformation from 3-D space onto the 2-D plane.

如果你想象經歷一個變換會是感到很不舒服的。

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

你們也可以有一個從2-維到1-維的變換。

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

1-維空間實際上只不過是一根數軸，

• One-dimensional space is really just the number line,

因此像這樣的變換把在2-維的矢量而輸出一些數字吧了。

• so transformation like this takes in 2-D vectors and spits out numbers.

想一下網格保持平行和均等

• Thinking about gridlines remaining parallel and evenly spaced

這點對在這裏發生的所有的坍縮而有點混亂不清的。

• is a little bit messy to all of the squishification happening here.

所以在這樣的情況下，對綫性意味著什麽的視覺上的理解

• So in this case, the visual understanding for what linearity means is that

是如果你有一根間隔均等點子的綫條，

• if you have a line of evenly spaced dots,

一旦它們被映射到數軸綫上去，它會保持間隔均等的。

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

這些變換中有一個是被記錄成一個1x2的矩陣的，

• One of these transformations is encoded with a 1-by-2 matrix,

兩個列中都只有一個項。

• each of whose two columns as just a single entry.

代表著單位矢量停下地方的兩個列

• The two columns represent where the basis vectors land

爾每個列都只有一個數字，這數字就是

• and each one of those columns requires just one number, the number that that basis vector

單位矢量所停在的坐標。

• landed on.

這實際上是一個和點積(dot product)密切相關有意義的變換的類型。

• This is actually a surprisingly meaningful type of transformation with close ties to

而我將要在下一個錄像來講。

• the dot product,

之前，我鼓勵你們自己來圍著這個想法玩味一下，

• and I'll be talking about that next video.

靜下心來想一個像矩陣乘法，和綫性方程組那些東西的意義。

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

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

• equations

• in the context of transformations between different dimensions.

• Have fun!

Hey, everyone!

B2 中高級 中文 美國腔 矩陣 變換 單位 空間 輸出 輸入

# 3blue1brown 線性代數精髓第8章（Nonsquare matrices as transformations between dimensions | Essence of linear algebra, chapter 8）

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tai 發佈於 2021 年 02 月 07 日