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。

But several commenters have asked about non-square matrices,

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!