## Best Calculator for Linear Algebra and Calculus

Many universities and colleges allow students to use advanced graphical calculators for exams in math courses, including linear algebra, calculus and engineering. Having the appropriate calculator can prove to be a huge advantage in terms of getting the correct answer and saving time. Even if the handwritten manual working is required, a good calculator can help to check the correctness of the answer.

## Best Calculator for Linear Algebra

The key calculations in linear algebra include calculating the inverse and determinant of matrices. Additional features that are useful in calculators include the ability to calculate eigenvalues, eigenvectors, and RREF (reduced row echelon form). These features can be great time savers (or at the minimum, useful tools for checking your answer), as matrix operations are often tedious and prone to human error.

Our first choice would be Texas Instruments TI-84 Plus CE Color Graphing Calculator, Black. This is one of the flagship calculators by Texas Instruments. This type of calculator is something like a smart phone, you would need to download free apps that will greatly magnify its power in solving Linear Algebra equations. Some of the linear algebra capabilities of the TI-84 Plus are:

• Calculate adjoint of a matrix
• Solve Simultaneous Equations
• Calculate dot product, cross product of vectors
• Calculate inverse of matrix (when matrix is invertible)
• Calculate LDU decomposition, Cholesky factorization
• Calculate Eigenvalues and plot the characteristic polynomial
• Compute the Frobenius norm of a matrix
• Perform Gram Schmidt orthonormalization
• Calculate null space (kernel) if a matrix
• and more!

This calculator is good enough up till senior undergraduate level or even graduate level, for computational intensive modules, such as computational physics or any module that requires linear algebra calculations.

For a more budget-friendly version, you may check out the Texas Instruments TI-84 Plus Graphing Calculator, Black which is essentially the black and white version of the TI-84 Plus CE Color.

Or for an even cheaper option, you could opt for the older version Texas Instruments TI-83 Plus Graphing Calculator which still supports the linear algebra apps.

Often, linear algebra courses include applications to differential equations as well, which overlaps with the next topic on Calculus.

Update: For the latest TI-85/86 models, there is a new “Matrix Mode” inbuilt into the calculator that can perform basic matrix operations (including inverse) and other advanced operations such as LU Decompositions, and finding eigenvalues/eigenvectors.

Note that paradoxically, the newer TI calculator models may be cheaper than the older ones.

## Best Calculator for Calculus

For Calculus, the important features that a calculator should have are the ability to perform numerical differentiation and integration. There are also some amazing apps for TI83/84 that can perform symbolic differentiation, that is, find the derivative of a function in symbols (not just the numerical value at certain values of x).

So once again, the TI-84 Plus CE Color Graphing Calculator, White is a good suggested calculator for Calculus modules.

If your budget (and course requirements) permit, you should definitely check out the Texas Instruments TI-Nspire CX II CAS Color Graphing Calculator with Student Software (PC/Mac). CAS stands for “computer algebra system”, which means that the calculator is capable of producing symbolic results rather than just numerical results.

The TI-Nspire™ CX II features a deSolve wizard function for reducing syntax errors in solving differential equations. The TI-Nspire CX can even do vector calculus. It is a very powerful smart calculator that is essentially a mini-computer.

## TI Calculator Alternatives

In my experience as a student and educator, TI (Texas Instruments) calculators dominate the market and are generally considered a good choice. But there are other viable alternatives that can also be considered good for linear algebra, calculus or engineering math.

A popular alternative is Casio calculators, for instance Casio fx-9750GII Graphing Calculator with icon based menu. Color white. An immediate advantage of Casio calculators is their low price.

The Casio fx-9750 (and related series) calculator is able to calculate determinant of matrix as well as basic matrix operations such as multiplying matrices, finding the inverse of matrix, etc.

As mentioned above, the main advantage of Casio graphical calculators is their low price. For advanced features, the TI calculators definitely trump over the Casio calculators.

Another alternative to TI calculators are the HP graphical calculators. Let us check out the most powerful HP calculator, the HP 50g Graphing Calculator.

The HP 50g Graphing Calculator has a “MatrixWriter form” to facilitate the entry of matrices. It is essentially like an Excel spreadsheet. Once the matrix is entered, there is a myriad of functions that can be applied, such as finding the determinant, trace, transpose, and rank of a matrix.

In terms of Calculus, the HP 50g can perform a wide range of calculus functions such as:

• Limits and derivatives
• Anti-derivatives and Integrals
• Calculate the Taylor/Maclaurin series of a function symbolically (up to 4-th order relative power, i.e., the difference between the highest and lowest power in the expansion is 4)
• Partial Derivatives (Multivariate Calculus)
• Multiple Integrals such as double integral
• Vector Calculus such as del, gradient, divergence, curl
• Differential Equations
• Fourier series

The list above is quite impressive! Overall, the HP 50g Graphical Calculator is a very strong competitor to the TI series graphical calculators.

## Best Calculator for Engineering

Engineers often need to use Linear Algebra, as well as Calculus. Hence, the best calculator for engineering often overlaps with the best calculator for linear algebra or calculus. Do check out our above reviews and pick the calculator that best suits your needs.

In general, we have the below summary.

Best Budget Calculator: Casio graphical calculators, such as Casio fx-9750GII Graphing Calculator with icon based menu. Color white.

Safe choice, all-round best calculator: TI Texas Instruments Calculator, for instance the iconic Texas Instruments TI-84 Plus CE Color Graphing Calculator, Black

Super Powerful Calculator (very advanced and comprehensive features): HP 50g Graphing Calculator

## When religion (Pope Francis) met science (Stephen Hawking)

Also interesting is that the founder of the “Big Bang” theory was a Catholic priest!

A Catholic priest, Fr. Georges Lemaître, is “Father of the Big Bang Theory”

## The meeting was cordial, in spite of their diametrically opposed views regarding the existence of God.

[A surprising encounter took place between the pope, who holds that God’s mercy is infinite, and a scientist who rejects the existence of God due to the infinity of the Universe. On December 2, Stephen Hawking participated in a conference in Rome on the Big Bang, today considered the most widely accepted theory for the origin of the universe—a theory proposed by the Catholic priest Georges Lemaître (1894-1966) in 1927. ]

Pope Francis and Stephen Hawking: Faith and reason reach out to each other

British scientist Stephen Hawking and Pope Francis met face to face on November 28 in the Vatican, just days after that renowned astrophysicist had confidently declared that asking “what was before the Big Bang” is meaningless, since it “is like asking what lies south of the South Pole.”

The meeting between these two charismatic figures, world-renowned representatives of the Catholic Faith (Francis) and of reason based on scientific knowledge (Hawking), was rare but cordial, although they have completely opposite stances regarding the idea of God.

Pope Francis considers God to be an infinitely merciful Father, greater than the Universe He created — the same Universe that Hawking has been studying for decades and which he believes arose from nothing, without divine intervention.

Hawking stated that “the discovery of the expansion of the Universe” — the “big bang” theory —  has been one of the “most important intellectual discoveries” of recent times. However, “With James Hartle from the University of California, I worked out what physical conditions the early universe must have, if space-time had no boundaries in the past. Our model became known as the ‘No-Boundary Proposal,’” said Hawking, during the Science and Sustainability conference organized by the Pontifical Academy of Sciences at the Casina Pio IV in the Vatican.

Although the Big Bang is a well-accepted theory among experts, and it is fundamental for the formation of galaxies and planets, there is still no convincing answer to what happened before the explosion, what caused it, and consequently the reason for existence. Nevertheless, the theory fits into a logical explanation of the universe. “The message of science is that we are the children of logic and not of chaos,” explained Italian nuclear physicist Antonino Zichichi, a colleague of Hawking, in an interview with Vatican Radio.

## Happy Pi Day!

Happy Pi Day to all readers of Mathtuition88.com!

Check out our previous posts on Pi:

Did you know, Pi day is also Einstein’s Birthday?

How to get Pi on Calculator – Without pressing the Pi Button

The Mystery of e^Pi-Pi (Very Mysterious Number)

Euler’s proof of Pi^2/6 (Basel Problem)

Category Theory: How to Make Pi

Pi hiding in prime regularities

Also, check out our cooking blog for the following Pie recipes!

Beef Pie

Strawberry Cheesecake Pie

Lemon Pie

Easy Banana Pie

Fudgy Fudge Pie

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Coupon Value: S$22 Minimum Spend: Nil Travel by: 30 September 2018 ## Stephen Hawking dies aged 76 RIP Stephen Hawking. The British physicist was known for his work with black holes and relativity, and wrote several popular science books including A Brief History of Time. “We are deeply saddened that our beloved father passed away today,” a family statement said. At the age of 22 Stephen Hawking was given only a few years to live after being diagnosed with a rare form of motor neurone disease. ## Researchers studying the Flynn effect have found that scores on IQ tests have actually improved over time. This is an interesting read for parents and students. Basically, IQ is a complex notion that is dependent on various factors, including environment and educational factors. Source: ThoughtCo You’ve probably heard someone lament the state of “kids today”: that current generations aren’t as smart as the ones that came before them. However, psychologists who study intelligence have found that there isn’t much support for this idea; instead, the opposite may actually be true. Researchers studying the Flynn effect have found that scores on IQ tests have actually improved over time. Below, we’ll review what the Flynn effect is, some possible explanations for it, and what it tells us about human intelligence. ### What Is the Flynn effect? The Flynn effect, first described in the 1980s by researcher James Flynn, refers to the finding that scores on IQ tests have increased in the past century. Researchers studying this effect have found wide support for this phenomenon. One research paper, published by psychologist Lisa Trahan and her colleagues, combined the results of other published studies (which included a total of over 14,000 participants) and found that IQ scores have indeed increased since the 1950s. Although researchers have documented some exceptions, IQ scores have generally increased over time. Trahan and her colleagues observed, “The existence of the Flynn effect is rarely disputed.” ### Why Does the Flynn Effect Happen? Researchers have put forward several theories to explain the Flynn effect. One explanation has to do with improvements in health and nutrition. For example, the past century has seen a decrease in smoking and alcohol use in pregnancy, discontinuation of the use of harmful lead paint, improvements in the prevention and treatment of infectious diseases, and improvements in nutrition. As Scott Barry Kaufman writes for Psychology Today, “The Flynn effect serves as a reminder that when we give people more opportunities to prosper, more people do prosper.” In other words, the Flynn effect could be partially due to the fact that, over the twentieth century, we’ve started addressing many of the public health issues that prevented people in earlier generations from reaching their full potential. Another explanation for the Flynn effect has to do with societal changes that have occurred in the past century as a result of the Industrial Revolution. In a TED talk, Flynn explains that the world today is “a world where we’ve had to develop new mental habits, new habits of mind.” Flynn has found that IQ scores have increased the most rapidly on questions that ask us to find similaritiesbetween different things, and more abstract types of problem solving — both of which are things that we need to do more of in the modern world. Several ideas have been put forward to explain why modern society might lead to higher scores on IQ tests. For example, today, many more of us have demanding, intellectually rigorous jobsSchoolshave also changed: whereas a test at school in the early 1900s might have been more focused on memorization, a recent test might be more likely to focus on explaining the reasons for something. Additionally, more people today are likely to finish high school and go on to college. Family sizes tend to be smaller, and it has been suggested that this may allow children to pick up on new vocabulary words while interacting with their parents. It’s even been suggested that the entertainment we consume is more complex today. Trying to understand and anticipate plot points in a favorite book or TV drama may actually be making us smarter. ## Singapore’s Only Live Turtle And Tortoise Museum Is Closing This March & Here’s How You Can Help I have visited the Turtle Museum around 2015/2016. It was a very impressive collection of turtles and tortoises, right in the middle of the Chinese Gardens. Even includes rare species like the Alligator Snapping Turtle. Quite sad to hear that they are closing down. Land price / rental in Singapore is notoriously expensive, so I can understand that having a space to house all those turtles is very hard in land-scarce Singapore. I would estimate that their rental would easily be at least S$10,000 a month for such a large space. They have kept the ticket prices low at just $5 for each person. Do help by donating (https://gogetfunding.com/turtles-need-a-home/), or even just sharing this news will help. Every share leads to more views and potentially more donors. Singapore’s Chinese Garden is known for its imperial styled pagodas and arch building, but within all that beauty lies another: Singapore’s very own Live Turtle & Tortoise Museum. A Guiness World Record Holder, the museum holds over 500 turtles and tortoises from 40 different species, ranging from the most dangerous alligator snapping turtle to the world’s 2nd largest tortoise species, the Giant Aldabra. All this is run by Connie Tan and a handful of workers, as part of her family’s legacy. Unfortunately, this marvel may soon no longer be – the museum is scheduled to be evicted from Chinese Gardens by 31st March. We headed down to understand what drives Connie to keep this museum alive against all odds. ## Poll Results: Which JC is the Best? 62 people have voted for the best JC in Singapore. (Poll is at: Which JC is the best in Singapore? At top place is Nanyang Junior College 12.9% (8 votes), tied with Raffles Institution 12.9% (8 votes). Visit the poll to see the full results! Some other interesting options voted are as follows: Other Answer Votes DKJC 1 Definitely not SR 1 ## Stolen A-Level scripts: 75 of 238 affected students will re-sit Chemistry Paper 3 Sincerely wishing good luck to those retaking their exams. It is certainly not easy, especially for boys who by that time will be in National Service. Also, for a fact-intensive course like Chemistry, with each passing month it gets harder since one’s memory would not be as fresh. SINGAPORE: Of the 238 students whose A-Level Chemistry scripts were stolen in the United Kingdom last year, 75 have opted to be re-examined for the affected paper, the Singapore Examinations and Assessment Board (SEAB) told Channel NewsAsia on Monday (Mar 12). The students, who received their A-Level results on Feb 23, are from Anderson JC, Anglo-Chinese JC (ACJC), Hwa Chong Institution (HCI) and Nanyang JC (NYJC). The stolen scripts were for Paper 3, which consists of free-response questions and carries a weightage of 35 per cent of the entire H2 Chemistry examination. The final A-Level grades of the affected students were derived through a projection. But given the “unique circumstances”, the affected students were given the option to re-sit the H2 Chemistry Paper 3 in April or November, and have the better of their two grades recorded in their A-Level result slip and certificate. Those who wished to register for the re-examination had to do so by Friday (Mar 9). SEAB said that of the 75, there were 28 students each from AJC and ACJC, seven from HCI and 12 from NYJC. Out of the total, 57 of the candidates registered for the re-examination in April, while 18 registered for the re-examination in November ## The magic (and math) of skating on thin ice without falling in Very interesting video. Do try to imagine the sound of the ice and check out the video to confirm your guess. Very surprising sound! It is a pity that not many countries have such ice to skate in. The main mathematical principle is Archimedes’ Principle: Congelation ice, while a solid form of water, does bend slightly and acts like an elastic plate buoyed by the water below. To Anje, it’s Archimedes buoyancy principle in action. “A body partially immersed in water is buoyed by a force equal to the weight of the water displaced by the body,” he said. Stepping onto an inch-and-a-half thick piece of lake ice — much less doing laps on it — is a no-go for most people. But for experienced Swedish skaters Henrik Trygg and Mårten Anje, few things top skating on thin ice. In December, still photographer Trygg filmed Anje skating on 1.8 inches of fresh ice on a lake outside Stockholm. The resulting mini-documentary— filled with the eerie, laser-like sounds of bending ice — went viral in February. One shot shows the ice, commonly called “black ice,” visibly bending under the skater’s weight. Which raised the question: Why doesn’t this thin frozen surface break? For the answer, we turned to Anje, a 35-year veteran of nordic skating whose day job is calculating risk. He is a mathematician and actuary at a consulting firm. ## Brand’s Essence of Chicken Effectiveness Students studying hard for exams may often feel mentally tired, though technically they are sitting on the chair most of the time. This is because the brain actually uses around 20% of the body’s energy despite the brain only being 2% of total body weight. Not to mention if the child is in a sports CCA, in addition to being mentally exhausted, he/she is also physically tired after all the training. Having energy after school has ended is very important, as 90% of the important revision takes place at home after school has ended. ## Is Essence of Chicken Effective There is quite a lot of scientific research on the effectiveness of chicken essence: Effectiveness of Essence of Chicken in Improving Cognitive Function in Young People Under Work-Related Stress Effects of chicken essence on recovery from mental fatigue in healthy males Chicken essence is basically condensed chicken soup, which contains a lot of vitamins and amino acids in easily absorbed form. This may be even better than eating vitamin pills which may not be absorbed well by the body. Brandsworld E-Store is the official online store of Brand’s, where you can buy Brand’s Chicken Essence online for your children, or even yourself. Other than Chicken Essence, Brand’s also sells other health products such as Birds Nest, and Collagen. ## Free Entry into Science Centre (In Conversation with … John Edmark) Students interested in the fusion of math, science and art may be interested to attend this wonderful event. Note: Pre-registration is needed for your free entry into the Science centre. 24 March 2018 10:30 – 12:00 Location: Maxwell Auditorium The Tinkering Studio @ Science Centre Singapore cordially invites you to interact with John Edmark, inventor, designer and artist who teaches design at Stanford University in Palo Alto, California. His most recent work is a series of animated sculptures called Blooms which endlessly unfolds and animates as it spins beneath a strobe light. Come join us for this talk where John will share his work with logarithmic spiral structures, Fibonacci numbers, and the Golden Ratio (Ф), and how it all led to his inventing Blooms. His masterful illusions are the results of a marriage between art and mathematics. RSVP: to Ms Jenny Leong by 18 March 2018. Pre-registration is required for your complimentary entry into the Science Centre. ## Wakin Chau 周华健 was a Math Major at University Something interesting that not many people know? Wakin Chau was born in a rice store owned by his family in Sai Ying Pun, Hong Kong. Growing up as the fourth son in his seven-member family, Wakin learned to play the guitar when he was about 13 years old. In 1979, he left for Taipei to major in mathematics at National Taiwan University. During his college years, he sang and played folk songs in local coffee shops. This activity was a tradition among NTU students, and it is how Wakin learned to sing in Mandarin, which would be key to his future as a music-maker in Taiwan. Source: Wikipedia Probably one of his most famous songs is “Friends”, 朋友. ## Latest SG Education News  Policies In Brief: Tuition fee subsidies The Straits Times Income eligibility criteria for the Mendaki Tertiary Tuition Fee Subsidy scheme will be revised upwards from August. … Students previously unable to take part in Outward Bound Singapore adventure training or overseas volunteer programmes organised by Youth Corps Singapore may get opportunities …  More Malay students from low-income families to benefit from revision of Tertiary Tuition Fee … Channel NewsAsia SINGAPORE: The Government will revise the income eligibility criteria for the Tertiary Tuition Fee Subsidy (TTFS) scheme to benefit more Malay students from low-income families. The TTFS scheme, introduced in 1991, covers tuition fees at tertiary institutions and benefits 11,000 students currently.  Subsidies for master’s courses to be removed or reduced for foreign, PR students: Ong Ye Kung TODAYonline SINGAPORE — From next year, foreign students will no longer receive subsidies for most of the vocation-based master’s degrees and post-graduate … For example, according to the tuition fee schedule for students admitted in the Nanyang Technological University last year, international students pay …  NUS to offer three new degrees in 2018, including a first in veterinary medicine TODAYonline After that, the students will spend another three years in Melbourne to complete their studies. While they will pay prevailing NUS and University of Melbourne tuition fees during their time in Singapore and Australia, Singaporean students can expect to pay about 15 per cent less in tuition fees for the DVM …  RGS student who had learning disability pays it forward The New Paper But her business was unsustainable as her students couldn’t pay tuition fees. I believe in … She then founded HER Planet Earth (HPE) last year to raise funds for underprivileged women affected by climate change and organisations such as Zero Waste SG and World Wide Fund Singapore. Today, the …  Polytechnic courses to be cut by 20%, more Normal (Academic) students to enter poly via through … TODAYonline If there isn’t any PFP, some students who have the potential to succeed in polytechnic cannot enter the school only because they didn’t do well in their O-Level exams,” said Mr Yap, who has an offer to study information systems at the Singapore Management University (SMU). “Imagine if there’s only a … ## Wanton Noodles As a Chinese born and grew up in northern part of China, I’m very familiar with noodles. Yet, its very difficult for me to cook southern pattern noodles well, like rice noodle, Hokkien mee. I tried wanton noodles before and it didn’t turn out well. I guess the main reason is that I didn’t pay much attention to the difference between southern noodles and northern noodles. Northern noodles are more hard and thicker. They need more time to be cooked. That is completely different from southern ones. I corrected my way of cooking wanton noodle this time. Noodle was bought from supermarket and so was the wanton wrap. 1. Prepare the wanton filling: Minced pork (100g), some shrimps ( around 10), 1 stalk spring onion, 1 small slice ginger, 3-4 water chestnut. Chop them together and let them be sticky to each other. Put chopped pork and shrimp in a… View original post 235 more words ## The Best Job: Mathematician The above survey was 2009 USA Job Market. By 2018, Number 1 Job is AI / Data Scientist in USA / China / Europe high-tech market, which still needs Mathematics. eg. • Linear Algebra (Matrix) • Calculus (eg. Gradient descent, …) • Bayesian Statistics (Probability), • Algebraic Topology (eg. Homological Algebra, etc), • Abstract Algebra (eg. Category Theory, etc…) • etc. View original post ## French Math “Coniques” : ellipses, paraboles, hyperboles. French Math is unique in treating these 3 conic curves: (ellipse, parabola, hyperbola), always starts from the first principle – a la the Cartesian Spirit “I think therefore I am” (我思故我在). “Catersian” Analytical Geometry was co-invented by two 17CE French mathematicians René Déscarte and Pierre de Fermat. Note: The “elliptic curve” is a powerful geometry tool used in Number Theory (proved the 350-year-old Fermat’s Last Theorem in 1994 by Andrew Wiles), also in the most advanced Encryption algorithm. View original post ## Chinese Competition (Total Prize of$20,000 SGD)

Quite a good deal if you are also interested in obtaining the HSK (Hànyǔ Shuǐpíng Kǎoshì) Grade 6 Qualification (HSK六级成绩报告), which will be useful if you study/work in China in the future.

Entry fee is S$90 for test takers at Singapore location. For Malaysia location, it is RM 250. By the way, Grade 6 is the highest grade in the HSK, designed for “learners who can easily understand any information communicated in Chinese and are capable of smoothly expressing themselves in written or oral form.”. This is open to both students and adults. Previous years youngest participant is 10 years old. Also, there is a “Singaporean and Malaysian Citizen” Category, together with an “International” Category. As we can see from the photos, there is a mixture of adults and children taking this test, there is no age limit. ## 重磅 | 比华语水平，赢丰厚奖金；总奖金2万新元，人人有机会！ 华语，一直是你的骄傲吗？ 曾经错过的，这一次全力把握； 曾经不甘的，这一次再决高低； 曾经获奖的，这一次重返荣耀！ 第三届国际华语水平大赛（新马地区） 欢迎各路华语水平高手勇敢参与！ “国际华语水平大赛（新马地区）”旨在提升新马各界人士学好华文的信心和自豪感。在新马两地社会各界的热烈响应下，大赛今年已傲然迈入第三个年头。前两届赛事共有逾千名来自各行各业的人士参与，其中不乏职场人士、华文教师、大学生、中学生，更有金发碧眼的外国人士。 ## 数学范畴化 Categorification https://www.zhihu.com/question/31823394/answer/53592663?from=timeline Key Points: • Category and Functor are above the underlying algebraic structures (Set, Group, Ring, Vector Spaces, etc), study the relations between these structures. • Early 19 CE mathematicians before “Category Theory” already knew there is 1:1 mapping between the Field Extension and Galois Group. Treat Structures and Relation between them (Functors) on equal f View original post ## Why Math is so Sexy? Cédric Villani (Field Medal 2010) mathematician becomes a deputé (Member of Parliament) in President Emmanuel Macron’s new party “En Marche” consisting of 90% non politicians. His new revolution in French Primary School Math Education is introducing “Singapore Math” : the 1960s Chinese secondary school 1 math (算术) modified by ex-南大 Prof Lee Peng Yee (李秉彝) with the Polya Problem Solving Method plus visual model diagram. View original post ## Six books that have shaped my mathematical worldview ## AI – DeepLearning – Machine Learning 3 Waves of AI Evolution: 1st Wave (1950s) : Alan Turing “The Father of AI” and his Princeton Prof Alonzo Church (Lambda Calculus). MIT Prof Malvin Minksy’s “Lisp” Functional Programming (a.k.a. Symbolic or Declarative) Language. 2nd Wave (1980s – 1990s) : Knowledge-Based Rule Engine Expert Systems. Failed because knowledge acquisition process is too difficult with limited rigid rules. 3rd Wave (2010s -): DeepLearning is the latest AI tool for Machine Learning, famous after 2016 “AlphaGo” game by a former Funan-center UK Kid Demis Hassabis (UK/Greek father & Singapore Chinese mom teacher) beat 2 “Go” World Champions (Korean Lee Sedol李世乭 and China 柯洁). Great Books Recommended 1. Learn Everything in 《Deep Learning》: • Math (eg. Gradient Descent – by French GrandMaster Cauchy 1847), • Linear Algebra (eg. Matrix, Eigen-decomposition), • Probability (eg. Bayesian, etc), • Key Deep Learning techniques. Note: Available at Singapore National… View original post 94 more words ## 《The Master Algorithm》 Main Point:One single “Master Algorithm” for all kinds of knowledge from chess to Go to Medicine to Stocks to …, only differ in ‘Data’ context, but the same One thinking “Mind”. 《The Master Algorithm》must have these 5 features: 1. Evolving-structure-based(ala genetic), 2. Learning parameters (connected) , 3. Probability-based (Baeysian), 4. Symbolic (knowledge representation as symbols), 5. Analogy-based (recognise similarities). • Symbolic like “equation” used in Physics: F = ma => F – ma = 0 => general equation “U (X) = 0” • Evolutionist: rules create new rules and discard old irrelevant rules, like the nature selection “survival the fittest “. • Overfitting Data: Bias vs Variance. Eg. A clock aways late by 1 hour (high B, low V); if it tells almost the right time but alternate erractically between fast & slow (low B high V) Induction is the inverse of Deduction: Deduction: A is a, (A = Socrates, a = human) if… View original post 200 more words ## The Pros & Cons of the French Elitist Grandes Ecoles The French Grandes Écoles System is characterized by one unique Ultra-Selective Exams – “Concours” (pronounced as Kongu) or 科举 (pronounced in Chinese dialect Fujian as “Kogu”) a la the 1,300- year Chinese Imperial Exams dated since 600AD till 1905. Napoléon Bonaparte had great admiration of the Chinese mandarin meritocractic selection system, he was influenced by the Jesuit priests who were mostly working in China coastal province Fujian, decided to implement “Concours” for his newly established military engineering college ”École Polytechnique” (aka ‘X’). Like any system, there are always two sides : the pros & the cons. Kogu served China well for 1300 years, producing top mandarins who ruled China with the most intelligent scholars through layers of selective exams from county (乡试选拔”秀才”) to province (省试选拔 “举人”) to the capital (京都 殿试选拔 “进士” – 前三名状元 /榜眼 / 探花). The Cons came from its Implementation “devils” – too focus on one… View original post 411 more words ## Reform in French Baccalaureat Current Baccalaureat Drawback: • Study too many subjects, • High Drop out New Reform: • Specialised on 4 major subjects – include compulsory French Literature (penultimate year before Bac) & Philosophy. • ala British A-level on 4 Advanced Subjects (eg. For Science stream: 1. Math, 2.Physics, 3.Chemistry /& Biology, 4.Economics) + 1 “Ordinary” Subject (English General Papers) + Project (Social / current affairs) https://www.economist.com/news/europe/21736539-reforming-french-education-will-not-be-easy-emmanuel-macron-wants-change-beloved View original post ## 英国留学生带你回忆被数学支配的恐惧 这位中国留英数学专业的小伙子有很好的口才: 数学之美, 他的数学遗憾。 华威大学（The University of Warwick)，famous for mathematics in UK. Key Points: Einstein用 Riemann Geometry 数学救了Newton 物理。 中国数学的没落: 自从明朝科举废除数学考试 3位大师救近代中国数学: 华罗庚, 陈省身, 苏步青 数学光明的未来: AI, Big Data, Cloud Computing 天才的崎岖道路: “扫地僧”张益唐 时代的”指数” exponential快脚步, 不要追风, 要一以贯之 : Prof. Andrew Wiles proved 350-year-old “Fermat’s Last Theorem” (FLT). [纠正]: Andrew Wiles 超过40岁, 没赶上Fields Medal, 只得个”奖励”。他看到椭圆(Ellipse)气球, 得到突破 FLT “工具”的灵感 – “Elliptic Curve” 。 数学是什么: 爱情, 艺术, 音乐, 科技 View original post ## Top 20 Math Books on Machine Learning / AI https://www.quora.com/If-you-could-buy-20-math-books-for-machine-learning-what-books-would-you-buy/answer/Sridhar-Mahadevan-6?share=75acc7bc&srid=oZzP The first 4 books (by Strang, Lang, etc) are the Masterpieces. • Linear Algebraby Strang. He writes math like few folks do, no endless paragraphs of definitions and theorems. He tells you why something is important. He wears his heart on his sleeve. If you want to spend a lifetime doing ML, sleep with this book under your pillow. Read it when you go to bed and wake up in the morning. Repeat to yourself: “eigen do it if I try.” Strang’s MIT OpenCourse: https://tomcircle.wordpress.com/2016/12/21/animation-linear-algebra/ https://tomcircle.wordpress.com/2016/02/28/best-linear-algebra-by-mit/ • Introduction toApplied Mathby Strang. You’ll need to understand differential equations at some point, even to understand the dynamics of deep learning models, so you’ll benefit from Strang’s tour de force of a survey through a vast landscape of ideas, from numerical analysis to Fourier transforms. • Algebraby Lang. This legendary Yale professor has written more “yellow jacketed” tomes… View original post 1,717 more words ## Royal Caribbean Promotion – Free Stuff to be won ## Royal Hunt QR Codes The first two tiers of rewards are Polar and Ya Kun$2 vouchers. The final tier is a chance to win the top prize (worth $15,000); a seven-night Mediterranean Cruise for two on board Royal Caribbean’s state-of-the-art Symphony of the Seas. The prize also includes return economy airfare to Barcelona, Spain. The Royal Hunt starts on March 2 and ends on March 20. Here is how you play: 1. Download the Royal Hunt app from the App Store or Google Playstore; 2. Activate the app and check the e-map. The e-map will show the locations of 20 MRT stations — Ang Mo Kio, Bedok, Bugis, Buona Vista, Chinatown, Choa Chu Kang, City Hall, Dhoby Ghaut, Harbourfront, Jurong East, MacPherson, Punggol, Tampines, Tanjong Pagar, Raffles Place, Serangoon, Stadium, Tiong Bahru, Toa Payoh and Woodlands; 3. Select any of the MRT stations indicated on the e-map and look for the Royal Hunt posters within each MRT station; 4. Scan the golden anchor QR code embedded within each Royal Hunt poster to drop the anchor onto each tile on the scorecard. After scanning each QR code, go to the next location to collect the subsequent Golden Anchors; 5. After collecting five Golden Anchors, you will be assigned a specific location to collect the sixth and final Golden Anchor; 6. After collecting the sixth Golden Anchor, you complete the scorecard and end the game. You will then be in the running to win the top prize. Six Golden Anchors must be collected to complete the scorecard. Hence, basically you need to complete and scan 6 QR codes out of the possible QR codes: Ang Mo Kio, Bedok, Bugis, Buona Vista, Chinatown, Choa Chu Kang, City Hall, Dhoby Ghaut, Harbourfront, Jurong East, MacPherson, Punggol, Tampines, Tanjong Pagar, Raffles Place, Serangoon, Stadium, Tiong Bahru, Toa Payoh and Woodlands QR codes. If you only want the first two tiers of rewards, 1 or 3 QR codes respectively instead of 6 QR codes will be sufficient. ## Royal Caribbean Promotions From time to time, Royal Caribbean is known to offer some of the following promotions: •$10 2nd Guest
• $10 offer on kids fare, balcony upgrades and second accompanying guest • 50% off 2nd guest • 3rd & 4th Guests Cruise FREE •$11 Suite upgrade
• 30% off all guests and Kids Sail Free offer

These promotions are usually for limited time only and interested customers are advised to watch out for such promotions by monitoring the official Royal Caribbean website.

Malcolm Gladwell is one of my favorite authors. He is very famous in the US, since his books have hit the bestsellers list in America multiple times.

I have written the following reviews on books/articles by Malcolm Gladwell:

Hence, one can see that it is a great honour to have Malcolm Gladwell to teach Masterclass. Sign up here: Malcolm Gladwell Teaches Writing.

## No homework, full-day school curriculum to help level playing field (Proposal, not implemented yet)

Full-day school is quite a drastic measure to combat tuition. Also, unless full-day means 7am to 7pm, it is unlikely to be different from the status quo.

Any parent with children in secondary school or JC is aware that school is already pretty much “full-day” as of today, from 7am to 5pm at the minimum on most days (including CCA). Hence, there is not much room to get more “full-day” than now. JC students are known to stay much later for CCA, probably some are already having schedules from 7am to 7pm, which more than qualifies as “full-day”.

Also, even if full-day school (say 7am to 7pm) is implemented, there seems nothing to stop students from having tuition during the weekends, or on weekdays 8pm-10pm.

Probably most students would not be too pleased at having a full-day school. If I were still a student, I would definitely be more stressed out by the full-day school. I would much rather have some homework but end school early. I would imagine teachers won’t be too happy too, full-day school for students means full-day school for teachers, since obviously some if not all teachers must stay back to supervise the students.

Most unhappy would be tutors, for obvious reasons. Probably if this is implemented, most tutors will have to change jobs. 😛

The underlying idea to level the playing field is good and makes sense though. Possibly make the full-day optional so that those who want to stay back and have the full-day can do so, those who want to leave can also do so.

Source: TodayOnline

SINGAPORE — To level the playing field for children from less advantaged socio-economic backgrounds, and break out of the country’s tuition culture, Nominated Member of Parliament Chia Yong Yong has suggested that all schools adopt a full-day curriculum.

That way, the children will complete their homework during school hours, and be able to spend more time on “push-frontier practicals” aimed at training them to become more comfortable in tackling problems and to grow an appetite for risk-taking. These qualities are essential traits for the current technological revolution, also known as Industry 4.0, she said.

In her Budget debate speech in Parliament on Wednesday (Feb 28), Ms Chia said the current academic model “runs the risk of not harnessing the potential of all our young people” who do not have access to enrichment and tuition classes. As a result, those from more advantaged socio-economic backgrounds who have access to these classes will outperform their peers.

Stressing that “every school is a good school, but not every home is equal”, the lawyer said the current system has been “abused” such that inequality continues to be perpetuated and deepened.

## NUS to offer veterinary medicine

Good news to those who want to be a vet! (Previously the usual option is to study in Australia). The veterinary medicine programme looks very rigorous, taking a total of 5.5 – 6.5 years, more than the duration of NUS medicine (5 years). At the end of the day, travel to Australia is still required though as it is a concurrent degree with University of Melbourne.

Source: Today

In a first for Singapore, the NUS will offer undergraduates a chance to study veterinary medicine through its new concurrent degree programme with the University of Melbourne. At present, only Ngee Ann and Temasek Polytechnics offer diploma courses respectively in veterinary bioscience and veterinary technology. The programme, which will take between 5.5 to 6.5 years to complete, will take in 10 students for a start, with the number to increase if demand is high.

They will spend 1.5 years at NUS to grow their foundation, followed by one-year abroad at the University of Melbourne to read modules in Year 1 of the Doctor of Veterinary Medicine (DVM). After that, the students will spend another three years in Melbourne to complete their studies. While they will pay prevailing NUS and University of Melbourne tuition fees during their time in Singapore and Australia, Singaporean students can expect to pay about 15 per cent less in tuition fees for the DVM portion of the programme as compared to the four-year course in Melbourne as an international student. They will graduate with an NUS bachelor’s degree in life sciences and a graduate-entry, professional veterinary science qualification from the University of Melbourne.

## Singapore Math Free Resources for Homeschool

Singapore Math is a popular resource for homeschooling children in the United States and many countries. Due to its challenging questions and unique methodology, Singapore Math prepares children for real math at higher levels.

We have curated some of the best Singapore Math Free Resources on the net:

1. https://thegoodgoatmomma.com/2014/10/17/free-singapore-math-curriculum-resources/
Has a good list of free resources including Curriculum and Tutorials.
2. https://singaporemathsource.com/resources/singapore-math-web-sites/
Some free online content to supplement Singapore Math.
3. https://www.freeeducationalresources.com/singaporemath/Singapore_Math.htm
This is an excellent treasure trove of free Singapore Math Worksheets (PDF).
4. https://www.onlinemathlearning.com/singapore-math.html
Many questions with worked solutions with Singapore Math Model method.
5. http://www.sgtestpaper.com/sgmath/
Includes questions modeled from real Singapore schools’ questions.

Do also read our most popular Singapore Math page to find out what is Singapore Math and what are its benefits.

## SOSD Flag Day (Help Stray Dogs)

If you are free do go support SOSD in their Flag day!

FLAG DAY 2018! FLAG COLLECTORS NEEDED!

DATE: 04 MARCH 2018 SUNDAY
VENUE: YOUTH PARK
TIME: 9am to 6pm

In a few months, we will be moving to our new shelter, and we wanted to begin this new chapter, with a very special Flag Day for 2018 – for the first time, we will be organising a combined Flag Day for SOSD , Animal Lovers League – ALL Authorized Page, and Oasis Second Chance Animal Shelter – OSCAS

https://sosdflagday2018.eventbrite.com

We will be bringing 40 to 50 dogs.You will find HDB-Approved dogs, young puppies, and adult dogs!

## Singapour : Les Maths Singapour- Une Methode Miracle

La remise du rapport Villani au Ministre de l’Education nationale Mr. Blanquer préconisant 21 mesures pour l’enseignement des mathématiques en France, a mis à l’honneur, par ricochet et par voix de presse, la méthode de Singapour pour l’apprentissage des maths.

https://lepetitjournal.com/singapour/les-maths-singapour-une-methode-miracle-224015

https://lepetitjournal.com/singapour/actualites/education-pourquoi-les-eleves-singapouriens-sont-ils-si-forts-en-maths-46043

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## Higher paying job than Doctor / Lawyer

We encourage top students to look beyond the traditional Singaporean jobs of Doctor / Lawyer as there are new emerging jobs that can equal or even surpass the pay of Doctor/Lawyer.

At the end of the day, do also consider your passion and aptitude, which may be more important than the salary. No point being stuck in a high paying job that you absolutely hate.

Do share this post with your children/relatives/classmates who may be choosing their courses after receiving their ‘A’ level results.

Source: Todayonline

SINGAPORE — A high-paying job as a doctor or lawyer has traditionally been the career path that many Singaporeans aspire to. But there is now a new kid on the block, with double degree graduates in business and computer science joining the ranks of top earners here.

According to the latest graduate employment survey released by three local universities on Monday (Feb 26), fresh graduates from Nanyang Technological University’s (NTU) business and computing science double degree programme commanded a median starting salary of S$5,000 last year, up from S$4,600 in 2016.

The median salary for the batch of 20 graduates matched that of their peers who graduated from the law and medicine faculties. They were also in demand with employers, as they recorded a 100 per cent overall employment rate.

Meanwhile, fresh computing science graduates were also among the highest paid last year. Those who graduated from this course in NTU got a median starting pay of S$3,850 last year, up from S$3,500 in 2016. Their counterparts from the National University of Singapore (NUS) received S$4,285 – S$285 more than in 2016.

However, rankings differed for 75th percentile salaries — the base salary of the top 25 per cent of the batch — as SMU-schooled lawyers emerged as top earners at S$5,840, compared to NUS doctors’ starting pays of S$5,305, and S$5,362 for NTU’s business and computer science graduates. Growth of starting salaries in law and medicine was tepid, however, as law graduates from NUS and SMU only received about S$100 and S\$150 more respectively last year, while NUS doctors banked in about the same amount as their seniors.

## 238 Singaporean A-level H2 Chemistry exam scripts stolen in the UK

Unbelievable. I think this should be the first time such an incident happen. Hope the affected students will get a fair grade despite this unfortunate incident.

Source: Yahoo

A parcel containing 238 scripts for the 2017 GCE A-Level H2 Chemistry Paper 3 was stolen from a courier in the United Kingdom, the Singapore Examinations & Assessment Board (SEAB) said Friday (23 February).

Students from four junior colleges – Anderson JC (58 scripts), Anglo-Chinese JC (60 scripts), Hwa Chong Institution (60 scripts) and Nanyang JC (60 scripts) – were affected, SEAB said in a statement. They comprised 3 per cent of the 8,843 school candidates who sat for the paper.

The theft occurred on 16 November 2017 while the parcel was in transit from Cambridge Assessment to the examiner. The case is under UK police investigations and the scripts have not been recovered.

## What to do if did not do well for A Levels

For those who did not do well for A Levels, do not be overly disheartened. Though it may seem like the end of the world, it is not, and there are other options available to you.

Firstly, check out this post by Digital Senior, where they discuss 7 alternative routes for those who did not do well for A Levels. The 7 alternative routes are: Polytechnic, SUSS (Full time/part time), Overseas universities, SIM-UOL, NTU/NUS part time degrees, SIT, and lastly retaking A levels.

Secondly, check out this post by Mothership, featuring 10 inspirational stories of Singaporeans who did not do well for A levels but persevered on to achieve success in their respective fields.

Thirdly, check out this Reddit post, where advice is given about failing A levels.

Also, do dispel the myth that failing GP will lead to zero chance of entering university. It is still possible to enter local university with failed GP, though certain courses like Law are ruled out.

If I failed my General Paper (GP) at the Singapore-Cambridge GCE A-Level exam, can I apply for admission?

Yes, you can apply for admission. However, if you failed GP, you will not qualify for certain courses such as Law which will require a GP grade of at least ‘B3’ (for old syllabus), ‘B’ (for new syllabus) or a good pass in Knowledge & Inquiry (KI) (for new syllabus).

Applicants who do not possess the necessary English Language qualification will be required to take the Qualifying English Test (QET), if offered admission. Those who fail to fulfill the QET requirement will not be allowed to graduate – click here for details.

Finally, just to share a quote about success and failure in general, not just A Levels:

## The results of the 2017 Singapore-Cambridge General Certificate of Education Advanced Level (GCE A-Level) examination will be released on Friday, 23 February 2018.

Good luck to all collecting their A Level results today!

Check out our post on BMAT Book Recommendations for NTU Medicine, and also Alternate Admission Route to NUS Computing.

1. The results of the 2017 Singapore-Cambridge General Certificate of Education Advanced Level (GCE A-Level) examination will be released on Friday, 23 February 2018. School candidates may collect their results from their respective schools from 2.30pm that day.

2. Private candidates will be notified of their results by post. The result slips will be mailed on 23 February 2018 to the postal address provided by the candidates during the registration period. Private candidates who have SingPass1accounts can also use their SingPass to obtain their results online via the internet Examination Results Release System (iERRS) on the Singapore Examinations and Assessment Board’s website (www.seab.gov.sg) from 2.30pm on 23 February 2018.

## 中一高华成语练习（一）Secondary 1 Higher Chinese Idioms Exercise I

Free Higher Chinese Worksheets

1，他乐于助人的事迹（        ），老师让我们向他学习。

2，十年树木，（         ）。

3，三人行必有我师，有问题要谦虚好学，（       ），才能更加进步。

4，这点帮助对这个不幸的家庭来说，只不过是（        ）。

5，倘若你没有经过努力，就想（       ），是不会取得真正的成功的。

6，真心喜欢一个人，往往会（        ），喜欢他的一切。

7，不管在生活中遇到什么困难，我们要有（         ）的精神。

8，爷爷（         ）在山上种树，终于让荒山披上了绿装。

9，你在著名华文作家的面前卖弄华文，岂不是（       ）。

10，看清楚！那只是根绳子，不是蛇！别（       ），自己吓自己。

11，不愿意付出努力就想获得成功，这不就是（        ）吗？

12，面对家乡（         ）的变化，他激动地一时说不出话来。

13，小伟的哥哥整天游手好闲, （        ）。

14，多读多写是学习写作的（       ）。

15，他做事一向（        ），有条不紊。

16，做事如果不从实际出发, （       ）, 就会造成不良的后果。

17，他为人（        ）, 怪不得能够在政商两界无往不利。

18，打仗主是要战胜敌人, （        ）, 敌人越骂我们狡猾, 越说明我们做得对极了。

19，他不时地长吁短叹, 话里话外总是透出一种（       ）， 以天下为己任的味道。

20，下车她才发现, 身上的钱包竟然（        ）了。

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## Heroic math teacher saved her students from Florida shooting

Heroic math teacher saved her students from Florida shooting by covering classroom door’s window, ordering kids to the floor and refusing to let anyone in… even the SWAT team. Read more: http://www.dailymail.co.uk/news/article-5399143/Heroic-teacher-saved-students-Florida-shooting.html#ixzz57i0Hcvi9

# Heroic math teacher saved her students from Florida shooting by covering classroom door’s window, ordering kids to the floor and refusing to let anyone in… even the SWAT team

• Mrs Viswanathan realised something was wrong after two fire alarms sounded
• Instead of letting pupils out her math class, she told them to duck in the corner
• Mrs V refused to let SWAT teams in, so they had to enter through the window
• A mother of a pupil in the class said Mrs V’s actions helped to save student’s lives

## Linear System of Differential Equations, Solutions, Phase Portrait Sketching

Solutions of Homogeneous Linear System of DE
$\displaystyle \mathbf{y}'=\mathbf{A}\mathbf{y}$
$\displaystyle \mathbf{y}(t)=\mathbf{v}e^{rt}$
where $r$ and $\mathbf{v}$ are eigenvalue and eigenvector for $\mathbf{A}$ respectively.

Superposition Principle
If $\mathbf{x_1}(t)$ and $\mathbf{x_2}(t)$ are two solutions to a homogeneous SDE $\mathbf{y'}=\mathbf{Ay}$, then $\displaystyle \mathbf{y}=c_1\mathbf{x_1}(t)+c_2\mathbf{x_2}(t)$ is also a solution for any scalars $c_1$, $c_2$.

Euler’s formula
$\displaystyle e^{i\theta}=\cos\theta+i\sin\theta$

General Solutions (Complex Eigenvalues)

1) Let $r_1=a+bi$ be an eigenvalue corresponding to eigenvector $\mathbf{v_1}$. (The eigenvectors are complex conjugates: $\mathbf{v_1,v_2}=\mathbf{p}\pm \mathbf{q} i$.)
2) Construct
$\displaystyle \mathbf{x}_\text{Re}(t)=e^{at}(\mathbf{p}\cos bt-\mathbf{q}\sin bt)$
$\displaystyle \mathbf{x}_\text{Im}(t)=e^{at}(\mathbf{p}\sin bt+\mathbf{q}\cos bt)$
3) The general solution is $\displaystyle \mathbf{y}=c_1\mathbf{x}_\text{Re}(t)+c_2\mathbf{x}_\text{Im}(t).$

## How to Sketch Phase Portrait

Probably the best video on how to sketch Phase Portrait:

## The Self-Driven Child (i.e. How to make your child motivated to study)

I think this book is a excellent read for Singaporean parents. Have your child ever seem to be too laid back and resist improvement, although he or she has great potential?

Often children in Singapore have to be “forced” to go to tuition to improve their grades, “forced” to study, “forced” to practice piano etc. Wouldn’t it be great if the child has self-motivation to willingly go for tuition / self-study? This change of mindset is explored in this book: “The Self-Driven Child”.

“Forcing” children to study works in the short term, producing temporary improved academic results, but in the long term it leads to psychological problems and even mental breakdowns in severe cases. Not to mention there is no joy of learning experienced by the child. The alternative solution is described in “The Self-Driven Child”.

“It is not an overstatement to say that this is one of the most radical and important books on raising healthy, resilient, purpose-driven kids.” – Madeline Levine, author of The Price of Privilege

“An invaluable resource for the thinking parent.” – Lisa Damour, author

Check out other recommended Motivational Books here.

## Characteristic Polynomial, Eigenvalues, Eigenvectors

Characteristic Polynomial, $\det(\lambda I-A)$
\begin{aligned} \lambda\ \text{is an eigenvalue of }A&\iff\det(\lambda I-A)=0\\ &\iff \lambda\ \text{is a root of the characteristic polynomial}. \end{aligned}

Eigenspace
The solution space of $(\lambda I-A)\mathbf{x}=0$ is called the eigenspace of $A$ associated with the eigenvalue $\lambda$. The eigenspace is denoted by $E_\lambda$.

Sum/Product of Eigenvalues
– The sum of all eigenvalues of $A$ (including repeated eigenvalues) is the same as $Tr(A)$ (trace of $A$, i.e. the sum of diagonal elements of $A$)
– The product of all eigenvalues of $A$ (including repeated eigenvalues) is the same as $\det(A)$.

## Finding Least Squares Solution Review and Others

Rotation Matrix

The rotation matrix
$\displaystyle R=\begin{pmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{pmatrix}$
rotates points in the $xy$-plane counterclockwise through an angle $\theta$ about the origin.

For example rotating the vector $(1,0)$ 45 degrees counterclockwise gives us:
$\displaystyle \begin{pmatrix} \cos 45^\circ & -\sin 45^\circ\\ \sin 45^\circ & \cos 45^\circ \end{pmatrix} \begin{pmatrix} 1\\ 0 \end{pmatrix} = \begin{pmatrix} \frac{\sqrt{2}}{2}\\ \frac{\sqrt{2}}{2} \end{pmatrix}.$

Finding Least Squares Solution

Given $Ax=b$ (inconsistent system), solve
$\displaystyle A^TAx=A^Tb$ instead to get a least squares solution of the original equation.

Projection

If we know a least squares solution $\mathbf{u}$ of $A\mathbf{x}=\mathbf{b}$, we can find the projection $\mathbf{p}$ of $\mathbf{b}$ onto the column space of $A$ by $\displaystyle \mathbf{p}=A\mathbf{u}.$

Dimension Theorem for Matrices (Also known as Rank-Nullity Theorem)

If $A$ is a matrix with $n$ columns, then $\displaystyle rank(A)+nullity(A)=n.$

($rank(A)$=number of pivot columns,

$nullity(A)$=number of non-pivot columns.)

Linear Independence and the Wronskian
A set of vector functions $\vec{f_1}(x), \dots, \vec{f_n}(x)$ from $\mathbb{R}$ to $\mathbb{R}^n$ is linearly independent in the interval $(\alpha,\beta)$ if $\displaystyle W[\vec{f_1}(x),\dots,\vec{f_n}(x)]\neq 0$ for at least one value of $x$ in the interval $(\alpha,\beta)$.