"Lisa: Well, where's my dad?

莉莎：那麽，我爸爸在哪裏？

Frink: Well, it should be obvious to even the most dimwitted individual who holds an advanced degree in hyperbolic topology that Homer Simpson has stumbled into

Frink：嗯，這甚至應該對最笨的有著連Homer Simpson也沾上些邊的抛物綫拓撲學高級學位都是很明顯的

... (dramatic pause) ...

...(一下子就停住了)...

the third dimension."

第三維空間。“

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

嗨老鄉們今天我有一個很簡短的錄像給你們，

just sort of a footnote between chapters.

只不過是在兩章之間的一個注解吧。

In the last two videos I talked about

在過去的兩個錄像裏我講過

linear transformations and matrices, but, I only showed the specific case of

綫性變換和矩陣，但是我只演示了

transformations that take two-dimensional vectors to other

把一個2維矩陣施加到一個

two-dimensional vectors.

2-維矢量的特殊情況。

In general throughout the series we'll work mainly

在這整個系列中一般我們主要就

in two dimensions.

用2-維的。

Mostly because it's easier to actually see on the screen and wrap your mind around,

大多數是因爲這更容易來真實地在屏幕

but, more importantly than that

上來看並給你有個概念，但更重要的是

once you get all the core ideas in two dimensions they carry over pretty

一旦你得到在2-維中所有的核心思想

seamlessly to higher dimensions.

你可以毫無接縫地運用到更高維的空間。

Nevertheless it's good to peak our heads outside of flatland now and then to...

不管怎樣這總是好的現在從扁平的2-維把我們的頭伸出來然後來...

you know see what it means to apply these ideas in more than just those two dimensions.

你知道的看看把這些想法用到那些高于2-維的空間是什麽意思。

For example, consider a linear transformation with three-dimensional vectors as inputs

例如，考慮一個綫性變換與一個3維的

and three-dimensional vectors as outputs.

矢量作爲輸入而3-維的矢量作爲輸出。

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

我們可以把這看作移動在以一個網格來代表的3-維空間中所有的點，

as represented by a grid, in such a way that keeps the grid lines parallel

以那樣一個方法使網格保持平行

and evenly spaced and which fixes the origin in place.

並間隔均等以及原點固定。

And just as with two dimensions, every point of space that we see moving around

就像在2維空間中一樣，我們看見在空間在移動著的點

is really just a proxy for a vector who has its tip at that point,

實際上只是一個矢量它的箭頭所在的點

and what we're really doing is thinking about input vectors

而我們真正在做的是在考慮一些輸入矢量

*moving over* to their corresponding outputs,

*搬到*它們相對應的輸出

and just as with two dimensions,

并且只是和在2-維的一樣，

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

這些變換中的一個完全是由基本矢量所去的來描述的。

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

但是現在，這裏我們通常使用的有三個標準的基本矢量：

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

在x-方向上的單位矢量，i-hat；

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

在y-方向上的單位矢量，j-hat；

and a new guy—the unit vector in the z-direction called k-hat.

而一個新的在z-方向上的單位矢量，叫做 k-hat。

In fact, I think it's easier to think about these transformations

事實上，我認爲通過只跟隨下面那些基本的矢量

by only following those basis vectors

來考慮這些變換是要更容易些

since, the for 3-D grid representing all points can get kind of messy

因爲，用3-維網格代表的所有的點可以變得有點混亂。

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

通過在背景中留一份原來的三個軸的副本，我們可以

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

來思考這三個基本矢量停下來的坐標。

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

記錄這三個矢量的坐標作爲一個3x3 矩陣的三個列。

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

這就給出一個矩陣只用了9個數字就完全地描述了這個變換。

As a simple example, consider, the transformation that rotate space

作爲一個簡單的例子，想一下，

90 degrees around the y-axis.

一個變換圍著y-軸轉90度

So that would mean that it takes i-hat

因此那會意味著它把i-hat

to the coordinates [0,0,-1] on the z-axis,

搬到z-軸上，坐標[0,0,-1]，

it doesn't move j-hat so it stays at the coordinates [0,1,0]

它沒有移動所以j-hat仍停在坐標[0,1,0]上

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

然後k-hat移到x-軸上在[1,0,0]上。

Those three sets of coordinates become the columns of a matrix

這三組坐標成爲一個矩陣的三個列

that describes that rotation transformation.

而矩陣描述了那個轉動的變換。

To see where vector with coordinates XYZ lands the reasoning is almost identical

來看有XYZ坐標的矢量停在什麽地方的

to what it was for two dimensions—each of those coordinates can be thought of

理由對2-維的那個幾乎是相同的--每個那些坐標可以想作

as instructions for how to scale

怎樣來對每個基本矢量乘以係數的指令

each basis vector so that they add together to get your vector.

以便來得到你的輸出矢量。

And the important part just like the 2-D case is that this scaling and adding process

而像在2-維情況下一樣重要的部分是乘以係數和加法的過程

works both before and after the transformation.

在變換的之前和之後都行的。

So, to see where your vector lands you multiply those coordinates

因此，要知道你的矢量停在什麽地方你

by the corresponding columns of the matrix and

乘上與矩陣相對應的列

then you add together the three results.

然後你把三個結果加在一起。

Multiplying two matrices is also similar

兩個矩陣相乘意思相似的

whenever you see two 3×3 matrices getting multiplied together

每當你看到兩個3x3矩陣相乘

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

你應該想象先施加由右面的那個所編碼了的變換

then applying the transformation encoded by the left one.

然後再施加由左面的那個所編碼了的變換。

It turns out that 3-D matrix multiplication is actually pretty

結果是3-維矩陣的乘法實際在像

important for fields like computer graphics and robotics—since things like

計算機圖像和機器人領域裏相當重要--

rotations in three dimensions can be pretty hard to describe, but,

因爲像在3-維中的轉動可以是很難來描述的，但是

they're easier to wrap your mind around if you can break them down as the composition

他們比較方便些在心裏能完全地來控制如果你可以把他們的合成分別地

of separate easier to think about rotations

分解開來來考慮各種轉動。

Performing this matrix multiplication numerically, is, once again pretty similar

用數字來做這個矩陣的乘法，

to the two-dimensional case.

又是和2-維的情況一樣。

In fact a good way to test your understanding of

事實上一個好辦法了測試你對前兩個

the last video would be to try to reason through what specifically this matrix

錄像的理解就是特別地來試試

multiplication should look like thinking closely about how it relates to the idea

這個矩陣乘法應該看上去像怎麽樣的

of applying two successive of transformations in space.

仔細考慮一下和怎樣用兩個在空間裏連續變換的想法的關係。

In the next video I'll start getting into the determinant.

在下一個錄像中我將來講行列式。