Name: 3blue1brown 線性代數精髓系列第9章 - Dot products and duality（Dot products and duality | Essence of linear algebra, chapter 9）
Uploaded: 2021-02-16T09:17:30.000Z
Duration: 14 min 12 s

關係子句

Traditionally, dot products or something that's introduced really early on in a linear algebra

傳統上，點積或者是在一個綫性代數課程

中很早就引進的東西，典型的是在一開始時。

因此我在這個系列中把它們推到那麽後面看起來有點奇怪。

So it might seem strange that I push them back this far in the series.

我這樣做是因爲有一種標準的方法來引入這個題目

I did this because there's a standard way to introduce the topic which

它只需要對矢量有一點基本的理解

requires nothing more than a basic understanding of vectors,

但對點積在數學中鎖扮演的角色的一種更全面理解

but a fuller understanding of the role the dot products play in math,

只能在綫性代數的幫助之下才做得到。

can only really be found under the light of linear transformations.

在那個之前，不過，讓我簡單的講一下

Before that, though, let me just briefly cover

the standard way that products are introduced.

這我設想至少有很多聼眾是看到過的。

Which I'm assuming is at least partially review for a number of viewers.

計算上來說，如果你有兩個相同維數的矢量

Numerically, if you have two vectors of the same dimension;

列出的數字的項數具有相同的長度，

所以矢量[1， 2]和 [3， 4]的點積，

矢量[6, 2, 8, 3]點積[1, 8, 5, 3]會是

The vector [6, 2, 8, 3] dotted with [1, 8, 5, 3] would be:

幸運的是，這種計算有一個真是很好的幾何解釋。

Luckily, this computation has a really nice geometric interpretation.

思考一下在2個矢量 v 和w之間的點積

To think about the dot product between two vectors v and w,

想象w投影到穿過原點和v 箭頭的一根綫上。

imagine projecting w onto the line that passes through the origin and the tip of v.

這個投影的長度乘以v 的長度，你就有了這 v・w 的點積

Multiplying the length of this projection by the length of v, you have the dot product

在這個w 的投影的方向和v 相反時

Except when this projection of w is pointing in the opposite direction from v,

而當兩個矢量都指向相同的方向的時候

that dot product will actually be negative.

So when two vectors are generally pointing in the same direction,

如果他們是互相垂直時，意思是

the projection of one onto the other is the 0 vector,

而如果他們大致是反向的，那麽它們的點積是負的。

現在，這個解釋注定是不對稱的，

And if they're pointing generally the opposite direction, their dot product is negative.

Now, this interpretation is weirdly asymmetric,

因此在我剛知道這點時，對次序無關我感到驚奇。

it treats the two vectors very differently,

so when I first learned this, I was surprised that order doesn't matter.

multiply the length of the projected v by the length of w

我是說，有沒有感到像是一個真正不同的過程？

對為什麽次序無關係有種直覺：

I mean, doesn't that feel like a really different process?

Here's the intuition for why order doesn't matter:

if v and w happened to have the same length,

完全就是v投影到w然後把投影的長度乘以

then multiplying the length of that projection by the length of v,

is a complete mirror image of projecting v onto w then multiplying the length of that

現在，如果你“放大”其中的一個，比方說v 放大2倍，

這樣他們就沒有一樣的長度了。

Now, if you “scale” one of them, say v by some constant like 2,

但是日我們來想請怎樣來解釋在這個

But let's think through how to interpret the dot product between this new vector 2v and

If you think of w is getting projected onto v

這是因爲在你‘放大’v2倍的時候，

This is because when you “scale” v by 2,

但是它把你在投影上去的矢量的長度加了一倍。

it doesn't change the length of the projection of w

但是在另一方面，假定你在想把v投影到w。

but it doubles the length of the vector that you're projecting onto.

好吧，這那種情況下，在我們被v乘以2的時候這投影的長度被‘放大了。

But, on the other hand, let's say you're thinking about v getting projected onto w.

這個你在投影上去的矢量之長度是一個常量。

Well, in that case, the length of the projection is the thing to get “scaled” when we multiply

所以縂的效果仍舊是這點積的加倍。

所以，即使在這個情況下對稱被破壞了，

The length of the vector that you're projecting onto stays constant.

這’放大‘ 對點積的值的效果，

So the overall effect is still to just double the dot product.

So, even though symmetry is broken in this case,

在我剛寫這些東西的時候，還有使我不清的一個大問題是

the effect that this “scaling” has on the value of the dot product, is the same

爲什麽搞這麽一個坐標對應的項

There's also one other big question that confused me when I first learned this stuff:

Why on earth does this numerical process of matching coordinates, multiplying pairs and

我們在這裏要把一個東西挖得深一些

它通常用‘雙重性’這個名字。

and also to do full justice to the significance of the dot product,

we need to unearth something a little bit deeper going on here

我需要用些時間講些從多維到單維，

這些些函數輸入一個2-維矢量而輸出某個數字。

I need to spend some time talking about linear transformations

但是綫性變換，當然，比你們的

from multiple dimensions to one dimension

用隨便一個2-維的輸入而輸出一個單維的函數更要嚴格些。

像我在第三章講到過的，在更高維數

These are functions that take in a 2D vector and spit out some number.

的變換中，有著一些正式的性質

But linear transformations are, of course,

much more restricted than your run-of-the-mill function with a 2D input and a 1D output.

但是為了不對我們最終目標分心我在這裏存心忽略那些，

As with transformations in higher dimensions,

而代之以集中在一個和所有正式的東西相當的視覺性質。

like the ones I talked about in chapter 3,

如果你有一根綫上面有間隔相等的點

there are some formal properties that make these functions linear.

But I'm going to purposely ignore those here so as to not distract from our end goal,

一個綫性變換將保持這些點間隔相等，

and instead focus on a certain visual property that's equivalent to all the formal stuff.

一旦它們到了輸出空間，那就是一個數軸。

否則的話，綫上的有些點就不均等了

那麽你的變換也就不是綫性的了。

a linear transformation will keep those dots evenly spaced,

正如你們以前已經看到過的一些例子，

once they land in the output space, which is the number line.

Otherwise, if there's some line of dots that gets unevenly spaced

它把i-hat和j-hat移到了什麽地方。

而這次，那些單位矢量分別停在一個數字上。

所以在我們記錄它們停下的地方作爲一個矩陣的列，

is completely determined by where it takes i-hat and j-hat

but this time, each one of those basis vectors just lands on a number.

讓我們通過一個例子對一個矢量施加一個這樣的變換的意味著什麽。

So when we record where they land as the columns of a matrix

比方是你有一個綫性變換它把i-hat停到1和j-hat 到 -2 .

each of those columns just has a single number.

跟蹤一個坐標為[4, 3]的矢量到什麽地方，

把這個矢量發成4 x i-hat + 3 x j-hat來考慮。