科学与数学betterexplained.com编辑精选
线性代数直观指南
An Intuitive Guide to Linear Algebra (2012)
2012 — 2022，它一次次回到首页
黑点是上首页（≥ 50 分）的那次投稿，空心点是没冲上去的投稿，点越大分数越高。
导读
BetterExplained 的线性代数直觉指南，用类比和几何图像（如把矩阵看作函数、用旋转地球仪解释特征向量）帮助读者建立对矩阵、行列式、特征值等概念的直观理解。文章多次登上 HN 首页，常被拿来与 3Blue1Brown 的视频系列比较。
为什么经久不衰：它把抽象的线性代数符号翻译成可想象的几何图像，多年来一直是许多人补上'直觉缺口'的第一站。
HN 评论精选
光看直觉讲解不够，必须经过做题的挣扎才能真正掌握
我最近刚学完 MIT OCW 的 Gilbert Strang 线性代数课。如果没有做题、读教材、看视频的挣扎过程，我根本学不会。像这篇文章和 3blue1brown 的内容固然值得称赞，但单靠它们无法把概念真正刻进脑子里。现在向量空间、范数、正交、秩、基这些概念几乎成了本能，让我学其他数学分支时不再觉得无从下手。因人而异，如果你能靠浓缩材料学会那很好，但我觉得真正的受益者其实是作者本人——他必须想清楚怎么把这些概念教给别人。
id_ris查看英文原文
I recently finished going through MIT OCW's linear algebra class from Gilbert Strang. Without the struggle of doing the assignments, reading the text, and watching the lectures, I don't think I would have ever learned the content. While content like this and that from 3blue1brown are commendable and useful, it simply would not have lodged the ideas into my head. Now that the ideas of things like vector spaces, norms, orthogonality, rank, basis, etc are nearly second nature, the concepts are useful as I study other branches of math which would feel impenetrable otherwise. YMMV, and if you can learn from condensed materials go for it, but I might be too dumb for it work lol. I think the real benefit accrues to the author who had to work out how to teach these concepts to others.
文章自称'直觉'却依赖记号背景知识，可能让读者更受挫
这个解释确实比我记忆中的高中数学清楚得多，但看公式示例时我还是很吃力。看到 A(x) = ax 时我不确定该怎么读：A 是接受 x 的函数吗？为什么等价于 a*x？把依赖额外上下文知识的东西称为'直觉'，轻则有点不诚实，重则有害——会让读者以为看不懂这种'直觉'材料是自己笨。不过实际例子（如斜率的 rise/run）确实清楚易懂，所以对记号也做个简单解释会更有帮助。
nearlynameless查看英文原文
While this explanation is certainly much clearer than what I remember of high school maths, I still have a pretty tough time following the formula examples. When I see A(x) = ax, I'm not entirely sure how to read it. Is A meant to be a function that accepts x? If so, why is the equivalent expression a * x? Is it supposed to be implied that function A also has some hidden value "a" that is going to be multiplied by the supplied value? Is this notation specific to multiplication, to this expression, or what? Positing that something is 'intuitive' when it depends so much on additional contextual knowledge seems ever so slightly disingenuous as best, and slightly harmful at worst; it can make the reader feel as though they must be dumb for not understanding this 'intuitive' material. I do acknowledge that this is linear algebra, and if one doesn't have a really solid grasp of notation of regular algebra it is likely to go over their heads, but the practical explanations (such as the slope rise/run example) are quite clear and relatively simple to follow; it follows that a simple explanation of the notation might be helpful too.
3Blue1Brown 的视频系列在直觉讲解上更胜一筹
这篇文章还行，但没有什么比 3B1B 在 YouTube 上的线性代数系列更直觉了（那个系列在 HN 上被发过几百次）。线性代数的本质是向量空间的线性变换，而这篇博文没有抓住这一点。
vecter查看英文原文
This is ok but nothing is as intuitive as 3B1B's series on YouTube that has been posted hundreds of times on HN [0]. Linear algebra is really about linear transformations of vector spaces, which is not captured in this blog post. [0] https://www.youtube.com/watch?v=fNk_zzaMoSs
文章填补了纯计算和纯理论两种极端教学之间的空白
大学线性代数的体验似乎分两种：要么是'线代/微分方程'合并课，不讲理论、只有大量实际例子和动辄十多页的习题集；要么是高度理论化的课，抽象到大一学生完全消化不了。我走的第一条路，虽然成绩不错但什么都没记住，后来上计算机图形课时一路哭着回家。这个链接看起来是个不错的中间地带。
eli_gottlieb查看英文原文
>We’re getting organized: inputs in vertical columns, operations in horizontal rows. OH! Bloody hell, so that's why we write it like that! >The eigenvector and eigenvalue represent the “axes” of the transformation. >Consider spinning a globe: every location faces a new direction, except the poles. >An “eigenvector” is an input that doesn’t change direction when it’s run through the matrix (it points “along the axis”). And although the direction doesn’t change, the size might. The eigenvalue is the amount the eigenvector is scaled up or down when going through the matrix. Very nice! I'd had an algebraic (ahaha) understanding of eigenvectors/values before, but hadn't a geometric intuition. Thank you. Now I can neatly imagine why the eigenvector is orthogonal/perpendicular to the "direction" of the transformation. Determinants, too. Excellent article.
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eli_gottlieb查看英文原文
>We’re getting organized: inputs in vertical columns, operations in horizontal rows. OH! Bloody hell, so that's why we write it like that! >The eigenvector and eigenvalue represent the “axes” of the transformation. >Consider spinning a globe: every location faces a new direction, except the poles. >An “eigenvector” is an input that doesn’t change direction when it’s run through the matrix (it points “along the axis”). And although the direction doesn’t change, the size might. The eigenvalue is the amount the eigenvector is scaled up or down when going through the matrix. Very nice! I'd had an algebraic (ahaha) understanding of eigenvectors/values before, but hadn't a geometric intuition. Thank you. Now I can neatly imagine why the eigenvector is orthogonal/perpendicular to the "direction" of the transformation. Determinants, too. Excellent article.
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