圖靈獎(jiǎng)獲得者哈明有一句名言：“計(jì)算的目的不在于數(shù)據(jù)，而在于洞察事物（“The purpose Of computing is insight，not numbers")。”此外，他還非常欣賞孔子的話：“學(xué)而時(shí)習(xí)之，不亦悅乎?！保堰@句話印在他著的《科學(xué)家和工程師用的數(shù)值方法》那本書的卷首作為座右銘（英文是To study，and when the occasion arises to what one has learned into practice is that not deeply satisfying?）?？v觀哈明的一生，他自己就是實(shí)踐這兩句話的一生。

通用科學(xué)計(jì)算軟件走入課堂和考場(chǎng)，走入科研和工作，成為學(xué)生、科學(xué)家和工程師的實(shí)驗(yàn)室，我們通過(guò)計(jì)算不僅僅獲得了結(jié)算結(jié)果和數(shù)據(jù)，還可以驗(yàn)證猜想和洞察規(guī)律。 我在以前的視頻里已經(jīng)舉了一些用北太天元求解高中數(shù)學(xué)問(wèn)題的例子，這里我再舉一個(gè)大學(xué)線性代數(shù)課上的例子:

?
?? 假設(shè)A為m*n的任意矩陣，B為n*m的任意矩陣。證明:
?? ??? ??? ?det（I + AB） = det (I + BA)
?? 這里I為相應(yīng)大小的單位矩陣。

有的教科書上是通過(guò)構(gòu)造一個(gè)特殊形式的塊狀矩陣而進(jìn)行證明的，這對(duì)很多同學(xué)來(lái)說(shuō)不是很直觀，會(huì)問(wèn)老師，你是如何想到構(gòu)造這個(gè)塊狀矩陣的？ 如果讓我們使用北太天元，我們可以用這些編程找找規(guī)律，可以以一種更加自然的方式找到證明思路。北太天元的代碼如下：

m = 3;
n = 5;
A = rand(m, n);
B = rand(n, m);
我們計(jì)算
det(eye(m) + A*B)
det(eye(n) + B*A)

我們會(huì)發(fā)現(xiàn)，的確 二者是相等的，然后我們想到 一個(gè)矩陣的行列式是它的所有特征值的乘積，那么 eye(m)+A*B 和 ey(n)+B*A 的特征值有什么規(guī)律，會(huì)相等嗎？ 兩個(gè)矩陣的行列式
相等，它們的特征值不一定非要有什么關(guān)系的，所以，我們可以用 eig 函數(shù)算算看

我們可以看到二者的不是1的特征值似乎有一個(gè)一一對(duì)應(yīng)關(guān)系， 進(jìn)一步似乎可以猜測(cè)
A*B 和 B*A 的非零特征值是相等的，可能連重?cái)?shù)也是相同的。
例如
????? A*B*x = lambda * x
其中 lambda 是 A*B 的非零特征值， 那么能不能證明 lambda 也是 B*A 的非零特征值呢?
在上面的等式的兩邊同時(shí)左乘以 B , 等
???? B*A*B*x = lambda * B*x
B*x 肯定不是0， 因此， 這就證明了 lambda 也是 B*A 的非零特征值。
我就不繼續(xù)了，請(qǐng)大家看看沿著這個(gè)思路是不是能夠把這個(gè)題目給證明出來(lái)， 然后還對(duì)這個(gè)題目有了更深的理解。 兩種做法都試試看，你覺得哪種方法更加自然一些 ？

除了哈明的名言，我還從網(wǎng)上搜集來(lái)了更多的名人名言，與大家共享。

?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


?數(shù)學(xué)語(yǔ)錄集，讓你獲得靈感和動(dòng)力


勵(lì)志名言是一種讓人興奮、感動(dòng)和分享激情的好方法。我們喜歡數(shù)學(xué)，所以我們的收藏集中在強(qiáng)大的數(shù)學(xué)語(yǔ)錄上。


有那么多了不起的數(shù)學(xué)數(shù)學(xué)家，他們擁有我們都可以學(xué)習(xí)的智慧。享受這個(gè)精心策劃的數(shù)學(xué)語(yǔ)錄集，它會(huì)定期更新。


在Twitter和Facebook上關(guān)注我們，獲取更多數(shù)學(xué)樂趣。請(qǐng)隨時(shí)在您的社交渠道上分享這些名言，并經(jīng)?；貋?lái)查看更多！



“給我一根足夠長(zhǎng)的杠桿和一個(gè)支點(diǎn)，我就能移動(dòng)世界?！?阿基米德



“計(jì)算的目的是洞見，而不是數(shù)字?！薄聿榈隆す?br>


“數(shù)學(xué)家為了美，從無(wú)窮多的設(shè)計(jì)中選擇了一種圖案，并將其付諸實(shí)踐。”-馬斯頓·莫爾斯



“電影是三種通用語(yǔ)言之一，另外兩種：數(shù)學(xué)和音樂。”-弗蘭克·卡普拉


“數(shù)學(xué)是一門語(yǔ)言?！?喬賽亞·威拉德·吉布斯


“很明顯，數(shù)學(xué)學(xué)習(xí)的主要目的必須是讓學(xué)生思考?！薄獥?br>

“數(shù)學(xué)是一門用簡(jiǎn)單的詞語(yǔ)表達(dá)困難想法的科學(xué)?！薄獝鄣氯A·卡斯納


“你可以在任何方面都有創(chuàng)造力——在數(shù)學(xué)、科學(xué)、工程、哲學(xué)方面——就像你在音樂、繪畫或舞蹈方面一樣?！薄狵en Robinson


“數(shù)學(xué)就像去健身房鍛煉大腦。它能磨礪你的頭腦。”-丹妮卡·麥凱拉



“數(shù)百萬(wàn)人看到蘋果掉了下來(lái)，但牛頓問(wèn)為什么?！薄{德·巴魯克



“正是數(shù)學(xué)本身常年的年輕性，使它與其他科學(xué)中令人不安的不朽性劃上了句號(hào)。”-埃里克·坦普爾-貝爾



“數(shù)學(xué)不允許虛偽和含糊?！?司湯達(dá)



“數(shù)學(xué)是文化的一個(gè)方面，也是算法的集合。”-卡爾·本杰明·博伊爾



“真理總是在簡(jiǎn)單中找到，而不是在事物的多樣性和混亂中找到?！?艾薩克·牛頓



“數(shù)學(xué)是藝術(shù)方面的，是對(duì)新節(jié)奏、秩序、設(shè)計(jì)、和聲的創(chuàng)造，而在知識(shí)方面，是對(duì)各種節(jié)奏、秩序的系統(tǒng)研究。”-威廉·L·沙夫



“我只能從事這項(xiàng)運(yùn)動(dòng)這么長(zhǎng)時(shí)間，我的余生都在學(xué)習(xí)數(shù)學(xué)?！?約翰·厄舍爾



“科學(xué)家很少?gòu)?qiáng)調(diào)我們無(wú)知的嚴(yán)重性?！?維拉·魯賓



“你必須花費(fèi)一些精力和精力才能看到數(shù)學(xué)中的美?！?瑪麗亞姆·米爾扎哈尼



“對(duì)我來(lái)說(shuō)，數(shù)學(xué)、計(jì)算機(jī)科學(xué)和藝術(shù)有著瘋狂的聯(lián)系。它們都是創(chuàng)造性的表達(dá)方式。”——塞巴斯蒂安·特龍



“數(shù)學(xué)的一個(gè)無(wú)限誘人的方面是，它最棘手的悖論可以發(fā)展成美麗的理論?！?菲利普·J·戴維斯



“關(guān)于數(shù)學(xué)，重要的是要記住不要害怕?！薄聿榈隆さ澜鹚?br>


“如果不能用數(shù)學(xué)方法證明，任何人類調(diào)查都不能被稱為真正的科學(xué)。”——達(dá)



“數(shù)學(xué)的研究，就像尼羅河一樣，始于細(xì)微，終于輝煌?！?查爾斯·凱勒布·科爾頓



“每一個(gè)新的發(fā)現(xiàn)都是數(shù)學(xué)形式的，因?yàn)槲覀儧]有其他指導(dǎo)?！?查爾斯·達(dá)爾文



“學(xué)習(xí)數(shù)學(xué)的唯一方法就是做數(shù)學(xué)。”-保羅·哈爾莫斯



“數(shù)學(xué)是科學(xué)的大門和鑰匙?！薄_杰·培根



“數(shù)學(xué)是唯一一個(gè)真理和美意義相同的地方?！?丹妮卡·麥凱拉




數(shù)學(xué)引號(hào)


“科學(xué)、工程和技術(shù)永遠(yuǎn)存在。數(shù)學(xué)永遠(yuǎn)存在。”——?jiǎng)P瑟琳·約翰遜



“你不必是一個(gè)數(shù)學(xué)家就能對(duì)數(shù)字有感覺。”——小約翰·福布斯·納什。



“數(shù)學(xué)是人類精神最美麗、最強(qiáng)大的創(chuàng)造?！薄固胤病ぐ图{赫



“數(shù)學(xué)表達(dá)了反映宇宙的價(jià)值觀，包括有序、平衡、和諧、邏輯和抽象美。”-迪帕克·喬普拉



“數(shù)字統(tǒng)治著宇宙?！?畢達(dá)哥拉斯



“我一直很喜歡數(shù)學(xué)。它是表達(dá)一個(gè)想法最精確、最簡(jiǎn)潔的方式?！薄{拉亞娜·穆爾蒂



在數(shù)學(xué)中，提出問(wèn)題的藝術(shù)必須比解決問(wèn)題更有價(jià)值



“如果人們不相信數(shù)學(xué)是簡(jiǎn)單的，那只是因?yàn)樗麄儧]有意識(shí)到生活有多復(fù)雜?！?約翰·馮·諾依曼



“我們找不到解決方案，并不意味著沒有解決方案。”-Andrew Wiles



“數(shù)學(xué)比較了各種各樣的現(xiàn)象，并發(fā)現(xiàn)了將它們聯(lián)系在一起的秘密類比?！?br>
-約瑟夫·傅立葉



“生活是一個(gè)數(shù)學(xué)方程式。為了獲得最大收益，你必須知道如何將消極轉(zhuǎn)化為積極。”-匿名者



“數(shù)學(xué)可能不會(huì)教會(huì)我們?cè)黾訍刍驕p少恨，但它給了我們希望，每個(gè)問(wèn)題都有解決方案?！?匿名者



“一個(gè)不是詩(shī)人的數(shù)學(xué)家永遠(yuǎn)不會(huì)是一個(gè)完整的數(shù)學(xué)家?！?卡爾·魏爾斯特拉斯



“純粹的數(shù)學(xué)家喜歡嘗試未解決的問(wèn)題——他們喜歡挑戰(zhàn)?！薄驳卖敗ね査?br>


“數(shù)學(xué)的本質(zhì)不是讓簡(jiǎn)單的事情變得復(fù)雜，而是讓復(fù)雜的事情變得簡(jiǎn)單?！?斯坦·古德爾



“哪里有數(shù)字，哪里就有美麗?！?普羅克洛斯



“為什么孩子們害怕數(shù)學(xué)？因?yàn)殄e(cuò)誤的方法。因?yàn)樗灰暈橐婚T學(xué)科。”-沙昆塔拉·德維



“不懂?dāng)?shù)學(xué)是理解世界的一個(gè)嚴(yán)重限制?！薄聿榈隆·費(fèi)曼



“什么是數(shù)學(xué)？它只是解決自然難題的系統(tǒng)努力?！?沙昆塔拉·德維



“我一直對(duì)用數(shù)學(xué)讓世界運(yùn)轉(zhuǎn)得更好感興趣?！?阿爾文·E·羅斯



“數(shù)學(xué)的本質(zhì)在于它的自由?！薄低袪?br>


“大自然是用數(shù)學(xué)語(yǔ)言書寫的。”-伽利略



“‘顯而易見’是數(shù)學(xué)中最危險(xiǎn)的詞?！?埃里克·坦普爾-貝爾



“數(shù)學(xué)不是關(guān)于數(shù)字、方程、計(jì)算或算法，而是關(guān)于理解?！薄けＡ_·瑟斯頓



“數(shù)學(xué)是理性的音樂?！薄材匪埂ぜs瑟夫·西爾維斯特



“不應(yīng)該有無(wú)聊的數(shù)學(xué)?！?Edsger Dijkstra



“成為一名數(shù)學(xué)家，靈魂上不可能沒有詩(shī)人?！?索菲亞·科瓦列夫斯卡婭



“深入研究任何事情，你都會(huì)發(fā)現(xiàn)數(shù)學(xué)?！?Dean Schlicter



“生活只有兩件事好，發(fā)現(xiàn)數(shù)學(xué)和教數(shù)學(xué)。”-西蒙·波松



“所有系統(tǒng)都以最佳方式運(yùn)行。你只需要弄清楚它們?cè)趦?yōu)化什么?！?亨利·波拉克



“數(shù)學(xué)是一個(gè)你可以做現(xiàn)實(shí)世界中做不到的事情的地方?！?Marcus du Sautoy