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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。

  • 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!

Hey, everyone!

嗨,大家好!

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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
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