北太天元提供洞见,而不仅仅是数据,以线性代数的一道证明题为例
图灵奖获得者哈明有一句名言:“计算的目的不在于数据,而在于洞察事物(“The purpose Of computing is insight,not numbers")。”此外,他还非常欣赏孔子的话:“学而时习之,不亦悦乎。”,把这句话印在他著的《科学家和工程师用的数值方法》那本书的卷首作为座右铭(英文是To study,and when the occasion arises to what one has learned into practice is that not deeply satisfying?)。纵观哈明的一生,他自己就是实践这两句话的一生。
通用科学计算软件走入课堂和考场,走入科研和工作,成为学生、科学家和工程师的实验室,我们通过计算不仅仅获得了结算结果和数据,还可以验证猜想和洞察规律。 我在以前的视频里已经举了一些用北太天元求解高中数学问题的例子,这里我再举一个大学线性代数课上的例子:
假设A为m*n的任意矩阵,B为n*m的任意矩阵。证明:
det(I + AB) = det (I + BA)
这里I为相应大小的单位矩阵。
有的教科书上是通过构造一个特殊形式的块状矩阵而进行证明的,这对很多同学来说不是很直观,会问老师,你是如何想到构造这个块状矩阵的? 如果让我们使用北太天元,我们可以用这些编程找找规律,可以以一种更加自然的方式找到证明思路。北太天元的代码如下:
m = 3;
n = 5;
A = rand(m, n);
B = rand(n, m);
我们计算
det(eye(m) + A*B)
det(eye(n) + B*A)

我们会发现,的确 二者是相等的,然后我们想到 一个矩阵的行列式是它的所有特征值的乘积,那么 eye(m)+A*B 和 ey(n)+B*A 的特征值有什么规律,会相等吗? 两个矩阵的行列式
相等,它们的特征值不一定非要有什么关系的,所以,我们可以用 eig 函数算算看

我们可以看到二者的不是1的特征值似乎有一个一一对应关系, 进一步似乎可以猜测
A*B 和 B*A 的非零特征值是相等的,可能连重数也是相同的。
例如
A*B*x = lambda * x
其中 lambda 是 A*B 的非零特征值, 那么能不能证明 lambda 也是 B*A 的非零特征值呢?
在上面的等式的两边同时左乘以 B , 等
B*A*B*x = lambda * B*x
B*x 肯定不是0, 因此, 这就证明了 lambda 也是 B*A 的非零特征值。
我就不继续了,请大家看看沿着这个思路是不是能够把这个题目给证明出来, 然后还对这个题目有了更深的理解。 两种做法都试试看,你觉得哪种方法更加自然一些 ?
除了哈明的名言,我还从网上搜集来了更多的名人名言,与大家共享。
A Collection of Math Quotes to Get You Inspired and Motivated
Motivational quotes are a great way to get excited, get moving, and share your passion. We love math, so our collection focuses on powerful math quotes.
There are so many amazing math mathematicians out there with nuggets of wisdom we can all learn from. Enjoy this curated collection of math quotes, which is updated regularly.
Follow us on Twitter and Facebook for more fun with math. Please feel free to share these quotes on your social channels, and check back often for more!
"Give me a lever long enough and a fulcrum on which to place it, and I shall move the world." - Archimedes
"The purpose of computing is insight, not numbers." - Richard Hamming
"Out of an infinity of designs a mathematician chooses one pattern for beauty's sake and pulls it down to earth.” - Marston Morse
“Film is one of the three universal languages, the other two: mathematics and music.” - Frank Capra
“Mathematics is a language.” - Josiah Willard Gibbs
“It is clear that the chief end of mathematical study must be to make the students think.” – John Wesley Young
“Mathematics is the science which uses easy words for hard ideas.” – Edward Kasner
“You can be creative in anything — in math, science, engineering, philosophy — as much as you can in music or in painting or in dance.” – Ken Robinson
“Math is like going to the gym for your brain. It sharpens your mind.” - Danica McKellar
“Millions saw the apple fall, but Newton asked why.” – Bernard Baruch
“It is the perennial youthfulness of mathematics itself which marks it off with a disconcerting immortality from the other sciences.” - Eric Temple Bell
“Mathematics allows for no hypocrisy and no vagueness.” - Stendhal
"Mathematics is as much an aspect of culture as it is a collection of algorithms." - Carl Benjamin Boyer
"Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things." - Isaac Newton
“Mathematics is on the artistic side, a creation of new rhythms, orders, designs, harmonies, and on the knowledge side, is a systematic study of various rhythms, orders.” - William L. Schaaf
"I can only play the sport for so long, I have mathematics for the rest of my life." - John Urschel
"Scientists too seldom stress the enormity of our ignorance." - Vera Rubin
"You have to spend some energy and effort to see the beauty in math." - Maryam Mirzakhani
“To me, mathematics, computer science, and the arts are insanely related. They’re all creative expressions.” – Sebastian Thrun
“One of the endlessly alluring aspects of mathematics is that its thorniest paradoxes have a way of blooming into beautiful theories.” -Philip J. Davis
“The important thing to remember about mathematics is not to be frightened.” – Richard Dawkins
“No human investigation can be called real science if it cannot be demonstrated mathematically.” – Leonardo Da Vinci
“The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.” - Charles Caleb Colton
"Every new body of discovery is mathematical in form, because there is no other guidance we can have." - Charles Darwin
"The only way to learn mathematics is to do mathematics." - Paul Halmos
“Mathematics is the gate and key to science.” - Roger Bacon
“Math is the only place where truth and beauty mean the same thing.” - Danica McKellar
math quotes
“There will always be science, engineering, and technology. And there will always, always be mathematics.” – Katherine Johnson
“You don’t have to be a mathematician to have a feel for numbers.” - John Forbes Nash, Jr.
“Mathematics is the most beautiful and most powerful creation of the human spirit.” - Stefan Banach
“Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty.” - Deepak Chopra
“Number rules the universe.” - Pythagoras
“I’ve always enjoyed mathematics. It is the most precise and concise way of expressing an idea.” - N.R. Narayana Murthy
“In mathematics, the art of proposing a question must be held of higher value than solving it.” - Georg Cantor
“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” - John von Neumann
"Just because we can’t find a solution, it doesn’t mean there isn’t one." - Andrew Wiles
"Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.”
- Joseph Fourier
"Life is a math equation. In order to gain the most, you have to know how to convert negatives into positives." - Anonymous
"Mathematics may not teach us to add love or subtract hate, but it gives us hope that every problem has a solution." - Anonymous
"A mathematician who is not also something of a poet will never be a complete mathematician." - Karl Weierstrass
"Pure mathematicians just love to try unsolved problems — they love a challenge." - Andrew Wiles
“The essence of math is not to make simple things complicated, but to make complicated things simple.” - Stan Gudder
"Wherever there is number, there is beauty." - Proclus
“Why do children dread mathematics? Because of the wrong approach. Because it is looked at as a subject.” - Shakuntala Devi
"To not know math is a severe limitation to understanding the world." - Richard P. Feynman
"What is mathematics? It is only a systematic effort of solving puzzles posed by nature." - Shakuntala Devi
“I’ve always been interested in using mathematics to make the world work better.” - Alvin E. Roth
"The essence of mathematics lies in its freedom." - Georg Cantor
“Nature is written in mathematical language.” - Galileo Galilei
“‘Obvious’ is the most dangerous word in mathematics.” - Eric Temple Bell
“Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding.” - William Paul Thurston
“Mathematics is the music of reason.” - James Joseph Sylvester
“There should be no such thing as boring mathematics.” - Edsger Dijkstra
“It is impossible to be a mathematician without being a poet in soul.” - Sofia Kovalevskaya
“Go down deep enough into anything and you will find mathematics.” - Dean Schlicter
"Life is good for only two things, discovering mathematics and teaching mathematics." - Simeon Poisson
“All systems operate optimally. You just have to figure out what they are optimizing over.” - Henry Pollak
"Mathematics is a place where you can do things which you can’t do in the real world." - Marcus du Sautoy
数学语录集,让你获得灵感和动力
励志名言是一种让人兴奋、感动和分享激情的好方法。我们喜欢数学,所以我们的收藏集中在强大的数学语录上。
有那么多了不起的数学数学家,他们拥有我们都可以学习的智慧。享受这个精心策划的数学语录集,它会定期更新。
在Twitter和Facebook上关注我们,获取更多数学乐趣。请随时在您的社交渠道上分享这些名言,并经常回来查看更多!
“给我一根足够长的杠杆和一个支点,我就能移动世界。”-阿基米德
“计算的目的是洞见,而不是数字。”——理查德·哈明
“数学家为了美,从无穷多的设计中选择了一种图案,并将其付诸实践。”-马斯顿·莫尔斯
“电影是三种通用语言之一,另外两种:数学和音乐。”-弗兰克·卡普拉
“数学是一门语言。”-乔赛亚·威拉德·吉布斯
“很明显,数学学习的主要目的必须是让学生思考。”——杨
“数学是一门用简单的词语表达困难想法的科学。”——爱德华·卡斯纳
“你可以在任何方面都有创造力——在数学、科学、工程、哲学方面——就像你在音乐、绘画或舞蹈方面一样。”——Ken Robinson
“数学就像去健身房锻炼大脑。它能磨砺你的头脑。”-丹妮卡·麦凯拉
“数百万人看到苹果掉了下来,但牛顿问为什么。”——伯纳德·巴鲁克
“正是数学本身常年的年轻性,使它与其他科学中令人不安的不朽性划上了句号。”-埃里克·坦普尔-贝尔
“数学不允许虚伪和含糊。”-司汤达
“数学是文化的一个方面,也是算法的集合。”-卡尔·本杰明·博伊尔
“真理总是在简单中找到,而不是在事物的多样性和混乱中找到。”-艾萨克·牛顿
“数学是艺术方面的,是对新节奏、秩序、设计、和声的创造,而在知识方面,是对各种节奏、秩序的系统研究。”-威廉·L·沙夫
“我只能从事这项运动这么长时间,我的余生都在学习数学。”-约翰·厄舍尔
“科学家很少强调我们无知的严重性。”-维拉·鲁宾
“你必须花费一些精力和精力才能看到数学中的美。”-玛丽亚姆·米尔扎哈尼
“对我来说,数学、计算机科学和艺术有着疯狂的联系。它们都是创造性的表达方式。”——塞巴斯蒂安·特龙
“数学的一个无限诱人的方面是,它最棘手的悖论可以发展成美丽的理论。”-菲利普·J·戴维斯
“关于数学,重要的是要记住不要害怕。”——理查德·道金斯
“如果不能用数学方法证明,任何人类调查都不能被称为真正的科学。”——达
“数学的研究,就像尼罗河一样,始于细微,终于辉煌。”-查尔斯·凯勒布·科尔顿
“每一个新的发现都是数学形式的,因为我们没有其他指导。”-查尔斯·达尔文
“学习数学的唯一方法就是做数学。”-保罗·哈尔莫斯
“数学是科学的大门和钥匙。”——罗杰·培根
“数学是唯一一个真理和美意义相同的地方。”-丹妮卡·麦凯拉
数学引号
“科学、工程和技术永远存在。数学永远存在。”——凯瑟琳·约翰逊
“你不必是一个数学家就能对数字有感觉。”——小约翰·福布斯·纳什。
“数学是人类精神最美丽、最强大的创造。”——斯特凡·巴纳赫
“数学表达了反映宇宙的价值观,包括有序、平衡、和谐、逻辑和抽象美。”-迪帕克·乔普拉
“数字统治着宇宙。”-毕达哥拉斯
“我一直很喜欢数学。它是表达一个想法最精确、最简洁的方式。”——纳拉亚娜·穆尔蒂
在数学中,提出问题的艺术必须比解决问题更有价值
“如果人们不相信数学是简单的,那只是因为他们没有意识到生活有多复杂。”-约翰·冯·诺依曼
“我们找不到解决方案,并不意味着没有解决方案。”-Andrew Wiles
“数学比较了各种各样的现象,并发现了将它们联系在一起的秘密类比。”
-约瑟夫·傅立叶
“生活是一个数学方程式。为了获得最大收益,你必须知道如何将消极转化为积极。”-匿名者
“数学可能不会教会我们增加爱或减少恨,但它给了我们希望,每个问题都有解决方案。”-匿名者
“一个不是诗人的数学家永远不会是一个完整的数学家。”-卡尔·魏尔斯特拉斯
“纯粹的数学家喜欢尝试未解决的问题——他们喜欢挑战。”——安德鲁·威尔斯
“数学的本质不是让简单的事情变得复杂,而是让复杂的事情变得简单。”-斯坦·古德尔
“哪里有数字,哪里就有美丽。”-普罗克洛斯
“为什么孩子们害怕数学?因为错误的方法。因为它被视为一门学科。”-沙昆塔拉·德维
“不懂数学是理解世界的一个严重限制。”——理查德·P·费曼
“什么是数学?它只是解决自然难题的系统努力。”-沙昆塔拉·德维
“我一直对用数学让世界运转得更好感兴趣。”-阿尔文·E·罗斯
“数学的本质在于它的自由。”——康托尔
“大自然是用数学语言书写的。”-伽利略
“‘显而易见’是数学中最危险的词。”-埃里克·坦普尔-贝尔
“数学不是关于数字、方程、计算或算法,而是关于理解。”——威廉·保罗·瑟斯顿
“数学是理性的音乐。”——詹姆斯·约瑟夫·西尔维斯特
“不应该有无聊的数学。”-Edsger Dijkstra
“成为一名数学家,灵魂上不可能没有诗人。”-索菲亚·科瓦列夫斯卡娅
“深入研究任何事情,你都会发现数学。”-Dean Schlicter
“生活只有两件事好,发现数学和教数学。”-西蒙·波松
“所有系统都以最佳方式运行。你只需要弄清楚它们在优化什么。”-亨利·波拉克
“数学是一个你可以做现实世界中做不到的事情的地方。”-Marcus du Sautoy