?此篇筆記是「線性代數(shù)筆記」系列的第6篇，記錄了「3Blue1Brown」的「Essence of linear algebra」系列課程的第9章內(nèi)容，討論了線性代數(shù)中「點(diǎn)積的概念」，強(qiáng)調(diào)了它們的「幾何解釋」以及「與線性變換的聯(lián)系」，以及「點(diǎn)積與投影之間的關(guān)系」；最后介紹了「對(duì)偶性」的概念，強(qiáng)調(diào)了「向量和線性變換之間的對(duì)應(yīng)關(guān)系」。
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Numerically, the 「dot product of two vectors with the same dimensions」 involves 「pairing up coordinates」, 「multiplying them」, and 「adding the results」.
在數(shù)值上，「兩個(gè)維數(shù)相同的向量的點(diǎn)積」涉及「配對(duì)坐標(biāo)」, 「相乘并將其結(jié)果相加」
Geometrically, it represents 「the projection of one vector onto another」.
在幾何上，它表示「一個(gè)向量在另一個(gè)向量上的投影」
「The order of dot product calculations does NOT matter」
「點(diǎn)積的計(jì)算順序無關(guān)緊要:」
If we take 「a line of evenly spaced dots」 and 「apply a linear transformation」
如果我們有一系列「等距分布于一條直線上的點(diǎn)」，然后進(jìn)行一個(gè)「線性變換」
In this case, 「each of basis vectors」 and just 「lands on a number」.
這一次，這些「基向量」 和 只「落在一個(gè)數(shù)上」
If a linear transformation that takes to and to , to follow where a vector ends up
「Place a number line diagonally in space」 somehow with the number 0 sitting at the origin, think of the 「two-dimensional vector」 , whose 「tips sit where the number 1 on the number line is」
「將一個(gè)數(shù)軸斜向放置在空間中」，保持 0 在原點(diǎn)，現(xiàn)在考慮這樣一個(gè)「二維向量」 , 它的「終點(diǎn)落在這條數(shù)軸的 1 上」
With this 「projection」, we just 「defined a linear transformation from 2D vectors to numbers」, so we're going to be able to 「find some kind of 1 x 2 matrix that describes that transformation」
根據(jù)這個(gè)「投影」, 我們定義了一個(gè)「從二維向量到數(shù)的線性變換」, 所以我們就能夠「找到描述這個(gè)變換的 1×2 矩陣」
we can 「reason through it」 with 「a line of symmetry」
我們可以通過「一條對(duì)稱線」進(jìn)行「推理」
So 「the entries of the 1 x 2 matrix」 describing the 「projection transformation」 are going to be the 「coordinates of 」.
所以「描述投影變換」的 「1×2 矩陣的兩列」就「分別是 的兩個(gè)坐標(biāo)」
「Computing this Projection Transformation」 for 「arbitrary vectors」 in space, which requires 「multiplying the Projection Matrix by those vectors」
空間中「任意向量」經(jīng)過「投影變換的結(jié)果」也就是「投影矩陣與這個(gè)向量相乘」
Loosely speaking, 「Duality」 refers to situations where you have a natural but surprising correspondence between 「two types of mathematical thing」, for the linear algebra case 粗略地說，對(duì)偶性指的是「兩種數(shù)學(xué)事物」之間「自然而又出乎意料的對(duì)應(yīng)關(guān)系」, 對(duì)于線性代數(shù)來說
??? Dot products are traditionally introduced early in linear algebra courses and involve multiplying coordinates and adding the results.
點(diǎn)積通常在線性代數(shù)課程的早期引入，涉及坐標(biāo)相乘然后將結(jié)果相加。?? The dot product can be geometrically interpreted as the length of the projection of one vector onto another.
幾何上，點(diǎn)積可以解釋為一個(gè)向量在另一個(gè)向量上的投影的長度。?? The dot product is positive when vectors point in the same direction, zero when they are perpendicular, and negative when they point in opposite directions.
當(dāng)向量指向相同方向時(shí)，點(diǎn)積是正的；當(dāng)它們垂直時(shí)為零；當(dāng)它們指向相反方向時(shí)為負(fù)。?? Order doesn't matter in dot product calculations, and the same result can be obtained by projecting the second vector onto the first.
在點(diǎn)積計(jì)算中，順序不重要，通過將第二個(gè)向量投影到第一個(gè)向量上可以得到相同的結(jié)果。?? Dot products have a connection to linear transformations, with a numerical computation similar to matrix-vector multiplication.
點(diǎn)積與線性變換有關(guān)，其數(shù)值計(jì)算類似于矩陣-向量乘法。?? Linear transformations from higher dimensions to one dimension can be described by 1x2 matrices, and the computation is equivalent to taking a dot product.
從高維到一維的線性變換可以用1x2矩陣描述，計(jì)算等價(jià)于進(jìn)行點(diǎn)積。?? Dot products demonstrate duality, which reveals a correspondence between vectors and linear transformations.
點(diǎn)積展示了對(duì)偶性，揭示了向量和線性變換之間的對(duì)應(yīng)關(guān)系。?? The dot product is a useful tool for understanding projections and vector relationships in geometry.
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點(diǎn)積是在幾何中理解投影和向量關(guān)系的有用工具。
[1] https://www.3blue1brown.com/lessons/dot-products
[2] https://github.com/3b1b/videos/blob/master/_2016/eola/chapter7.py
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