What Is Linear Algebra and Why Is It Important for CSIR NET, SET, IIT JAM & PG Exams?

 

Linear Algebra is one of the core subjects in mathematics that influences almost every major competitive exam. Its concepts solidify a strong foundation for advanced topics, which will help you develop strong analytical and problem-solving skills, without being left in the dark. This core subject carries high weightage and appears consistently every year in competitive exams such as CSIR NET, SET, IIT JAM, and various PG Entrance examinations. Since the questions from this subject are based on well-defined concepts and established techniques, it can be leveraged for scoring more marks.

What Is Linear Algebra?

Linear Algebra is the domain of mathematics that deals with the concepts of vectors, matrices, and systems of linear equations. This mathematics domain is concerned with the combination, scaling, and transformation of quantities. Linear Algebra is the means by which patterns can be understood, problems can be solved, or real-world situations can be modelled.

The conceptual basics of Linear Algebra functions as a foundation for advanced level of mathematics, making it a compulsory first step in the comprehensive learning of maths for competitive exams.

•        Matrices - Though a matrix is formally defined as a rectangular array of numbers, its significance goes far beyond this definition. In applications, a matrix represents much more: it can describe a transformation, encode an image, organize complex data, or model real-world systems. Thus, matrices serve as powerful tools that connect abstract mathematics with practical applications.

•     Vector Spaces - A vector space is a set of vectors on which you can perform addition between vectors and scalar multiplication while remaining within the space. They are the basis through which concepts of direction, dimension, and linear combinations are handled.

•        Linear Transformations - A linear transformation can be viewed as a function that takes vectors in one vector space and transforms them into vectors in another vector space in a manner that preserves both addition and scalar multiplication. It's the operation that combines vectors in a similar process that preserves their linear nature

•        Eigen values & Eigenvectors are unique vectors that do not change direction when acted upon by a linear transformation, while eigen values are the scalar multiples to which they are scaled. Both terms are very essential in solving equations and simplifying problems.

What is Linear Algebra’s relevance in Competitive Exams

1. Linear Algebra in CSIR NET Mathematics

•        In CSIR NET Mathematics, Linear Algebra problems are commonly found in Part B and Part C. The understanding of concepts, in this case, requires clarity in concepts because a question would, in most cases, need logical reasoning as opposed to mere calculation.

•        MathLab’s online linear algebra coaching for CSIR Net covers the entire Part B & C in detail, to ensure that aspirants never stumble when they come across its complexities.

2. Linear Algebra in IIT JAM Mathematics

•        In IIT JAM Mathematics, Linear Algebra is given high weightage, where the problems are often from matrices, system of equations, vector spaces, and eigenvalues. As the problems are generally moulded around concepts and are structured properly, understanding Linear Algebra proves to be crucial for cracking Rank Improvement attempts.

•        MathLab institute is one of the top coaching centres in Kerala that provides the best Linear algebra preparation for IIT JAM.

3. Linear Algebra in SET & PG Entrance Exams

•        Linear Algebra is one of the most predictable portions for SET or PG entrance exams because it's either repeated or pattern based, making targeted preparation highly effective. Performance in this subject may be a make-or-break factor in cutting off and granting entry to college.

•        Linear Algebra coaching for PG entrance exams makes it easy for students to cover the entire syllabus with proper guidance, seamless time management, and proper revision.

Most Important Linear Algebra Topics for Competitive Exams

The Best way to learn linear algebra is to focus on its key topics that will most likely appear for the competitive exams -Systems of Linear Equations - Learn to solve linear equations with matrices, elimination, and determinants—a well-covered topic.

•        Rank of a Matrix - Knowing the rank can be used to assess the consistency and solvability of linear equations and to assess the nature of the image space corresponds to a linear transformation.

•        Vector Spaces & Subspaces - These lay the basis for Linear Algebra and enable one to comprehend dimensions, linear combinations, and linear independence.

•        Basis and Dimension - The basis vectors address the units of the structure of the vector space, whereas the dimension focuses on the number of vectors required.

•        Linear Transformations: Research the mapping of vectors from one space to another, along with related matrix notation.

•        Eigenvalues, Eigenvectors & Diagonalization - These ideas make linear transformations easy to understand, solve equations in calculus, and thus apply equations in general.

•        Inner Product Spaces: Inner product is a way to introduce angles, lengths, orthogonality, and projections, which arise in many complex mathematical exercises.

Why Linear Algebra Is a High-Scoring Subject 

Linear Algebra is a core subject that students must pay more attention to because of its high scoring potential with focused preparation. Most problems in this area are logic problems requiring students to understand and arrive at solutions through reasoning rather than memorization since fewer fundamental formulas are used. This makes revision easy.

Another major advantage is exam predictability. Topics like matrices, vector spaces, linear transformations, and eigenvalues appear consistently across competitive exams, often with familiar patterns. This allows students to prepare strategically, practice standard problems, and score confidently.

Role of Linear Algebra in Other Mathematics Subjects

•        Functional Analysis - Vector spaces and linear operators are the very basics of Functional Analysis, and knowledge about topics such as inner product spaces and linear transformations can help in understanding infinite-dimensional spaces & advanced theorems.

•        Differential Equations - A lot of Differential Equation problems involve matrices, eigenvalues, and eigenvectors, which come from Linear Algebra. Having good mathematical foundations makes it easy for you to solve linear Differential Equation sets effectively.

•        Numerical Methods - Numerical methods make extensive use of the topic of Linear Algebra relating to the solution of equations, approximations, and iteration. With proper preparation for Matrices and determinants for competitive exams, aspirants can acquire knowledge that enables them to solve equations using algorithms with correct error estimates.

How to Prepare Linear Algebra for CSIR NET, SET & IIT JAM

•        Build basic concepts - The first step in Linear Algebra preparations for competitive exams is to strengthen your foundational knowledge of the mathematical subject. Fixing a solid base will set your preparations off on a strong footing - that’s why linear algebra tutors always spearhead fundamentals before progression.

•        Practice standard problems - Questions on matrices, determinants, vector spaces, linear transformations, and eigenvalues are repeatedly asked in CSIR NET, SET, and IIT JAM exams, which is why it's important to practice them regularly, to be better prepared with the exam, grasp patterns, solve problems efficiently, and be able to complete it within the allotted time.

•        Solve previous year questions - Understanding exam patterns is a pre-requisite for better preparations, which makes previous year question papers an essential resource tool that will help you understand weightage, repetitive questions, and spot weak links.

•        Take topic-wise tests - Understanding your strengths, weaknesses, and pinpointing those areas you haven’t covered is a fundamental stage of your preparations. Taking topic-wise tests and mock exams will help you map out your preparation progress, your weak links, and where you need to allocate time for improvement.

126 Best New Year’s Resolution Ideas for Students

Introduction – Why New Year Resolutions Matter for Students

New Year always feels like a fresh start. It’s the perfect time to reset your mindset, drop bad habits, and pick up new ones that can actually change your life. For students, resolutions aren’t just about hitting the gym or eating less junk food (though that helps too). They’re about building habits that will set you up for success in studies, health, career, and personal growth.

The truth is, resolutions can either become forgotten wishes or real lifestyle changes. The difference? How you set them, and how consistent you are. This post is your ultimate list of 126 New Year’s resolution ideas for students, with tips to make them stick.

How to Set Resolutions That Actually Work

We’ve all made that one resolution like “I’ll study harder this year” only to forget it by February. The problem isn’t motivation—it’s structure.

Here’s how to make resolutions stick:

Use SMART Goals

• Specific → “I’ll read 2 books a month” (not just “read more”).
• Measurable → Track your progress with apps or journals.
• Achievable → Don’t set crazy goals like 10 hours of study daily.
• Relevant → Align goals with what actually matters to you.
• Time-bound → Add deadlines so you stay consistent.

Habit Stacking: Attach a new habit to an existing one.

• Example: “After brushing my teeth at night, I’ll journal for 5 minutes.”

Start Small: Big results come from small, consistent steps.

New Year Resolution Categories

Here’s where the fun begins. 126 ideas are broken into categories so you can pick the ones that fit your lifestyle.

Health & Fitness Resolutions

1. Sleep 7–8 hours regularly.
2. Drink at least 8 glasses of water daily.
3. Exercise 3x per week (even short workouts count).
4. Take the stairs instead of elevators.
5. Limit junk food to weekends.
6. Try meal prepping once a week.
7. Cut down on sugary drinks.
8. Add one fruit to your diet daily.
9. Practice stretching or yoga every morning.
10. Go for a daily 20-minute walk.
11. Replace snacks with nuts or healthy alternatives.
12. Join a fitness class (Zumba, pilates, kickboxing).
13. Try intermittent fasting if it suits your body.
14. Meditate for 5 minutes daily.
15. Track your steps with an app or smartwatch.
16. Focus on posture during study sessions.
17. Take regular screen breaks.
18. Try deep breathing exercises.
19. Limit caffeine intake.

 20.Get annual health check-ups.

Personal Growth Resolutions

21. Read at least one book per month.
22. Start journaling (gratitude, reflection, or bullet journal).
23. Wake up earlier. 
24. Limit screen time before bed.
25. Learn a new language.
26. Practice public speaking.
27. Improve handwriting or note-taking style.
28. Create a morning routine that energizes you.
29. Write affirmations daily.
30. Develop a growth mindset (believe you can improve).
31. Listen to podcasts while commuting.
32. Reduce procrastination by using timers.
33. Start a side creative project.
34. Set weekly personal challenges.
35. Learn meditation or mindfulness.
36. Keep a “proud moments” journal.
37. Practice positive self-talk.
38. Spend more time outdoors.
39. Limit negative social media content.

  40.Learn to say “no” when necessary.

Career & Education Resolutions

41. Revamp your resume or LinkedIn profile.
42. Take an online certification course.
43. Improve typing speed.
44. Learn Excel, PowerPoint, or coding.
45. Create a study schedule and stick to it.
46. Revise notes weekly instead of last-minute cramming.
47. Join a career workshop.
48. Attend university networking events.
49. Learn a freelancing skill (graphic design, writing, coding).
50. Start building a portfolio.
51. Get an internship or part-time job.
52. Build strong relationships with professors.
53. Form or join a study group.
54. Start a blog or YouTube channel on your field.
55. Take practice tests regularly.
56. Explore scholarships and grants.
57. Improve presentation skills.
58. Learn research skills (databases, citation tools).
59. Shadow someone in your dream career.

  60.Work on time management techniques.

Financial Goals Resolutions

61. Start budgeting monthly.
62. Track your expenses daily.
63. Open a savings account.
64. Save at least 10% of pocket money/earnings. 
65. Avoid impulse shopping.
66. Learn basic investing(mutual funds, stocks).
67. Build an emergency fund.
68. Cook at home instead of ordering food daily.
69. Try “no-spend” days.
70. Sell old stuff you don’t use.
71. Start a small side hustle.
72. Learn about credit scores.
73. Apply for student discounts whenever possible.
74. Limit fast fashion shopping.
75. Buy second-hand textbooks.
76. Use cash instead of cards for better control.
77. Avoid unnecessary subscriptions.
78. Set savings goals for travel or gadgets.
79. Learn digital skills to earn online. 

  80.Read a personal finance book.

Relationships & Social Life Resolutions

81. Spend quality time with family weekly.
82. Call your grandparents more often.
83. Make time for close friends.
84. Limit toxic friendships.
85. Join a student club or society.
86. Plan monthly hangouts with friends.
87. Surprise a friend with a small gift.
88. Celebrate birthdays meaningfully.
89. Network with peers from your field.
90. Do random acts of kindness.
91. Volunteer for a cause.
92. Compliment people genuinely.
93. Listen more, talk less.
94. Plan a road trip with friends.
95. Write letters instead of just texts.
96. Start family traditions.
97. Share achievements with loved ones.
98. Forgive someone and move on.
99. Create boundaries in relationships.

 100. Support your friends’ goals.

Lifestyle & Fun Resolutions

101. Travel to a new city.
102. Try a new hobby (painting, photography, baking).
103. Start gardening (even small plants count).
104. Organize your study/workspace aesthetically.
105. Plan a monthly “me day.”
106. Create a vision board.
107. Watch documentaries instead of random shows.
108. Try digital detox weekends.
109. Start a collection (books, postcards, journals).
110. Explore local cultural events.
111. Learn cooking your favorite dish.
112. Try bullet journaling.
113. Visit a library or bookstore regularly.
114. Plan a picnic with friends.
115. Redecorate your room.
116. Take aesthetic photos for memories.
117. Write bucket lists.
118. Try DIY crafts.
119. Practice gratitude daily.

120. Watch sunrise or sunset more often.

126 New Year Resolution Ideas (Quick-Glance List)

To save your time, here’s the full mega checklist for you:

Health & Fitness (1–20)
Personal Growth (21–40)
Career & Education (41–60)
Financial Goals (61–80)
Relationships & Social Life (81–100)
Lifestyle & Fun (101–120)

To complete 126: Here are 6 extra “All-Rounder Resolutions” you can try:
121. Practice time blocking for productivity. 
122. Start reading 10 pages daily.
123. Build a morning routine.
124. Do one act of kindness every day.
125. Journal your progress monthly.
126. Believe in yourself, no matter what.

Tips to Stick With Your Resolutions

• Use a habit tracker app or journal (tick marks feel so satisfying ).
• Get an accountability partner (a friend with similar goals).
• Do monthly check-ins (don’t wait till December to review).
• Reward yourself for small wins.
• Be flexible—if one resolution isn’t working, tweak it instead of quitting.

Conclusion – Make This Year Count

New Year’s resolutions aren’t about becoming a completely different person overnight. They’re about building small, powerful habits that turn into lifestyle changes.

As a student, you have the best chance to use this time to grow—not just academically, but personally, financially, and socially. Start with a few resolutions from this list, stay consistent, and trust me—you’ll look back at the end of the year amazed at how far you’ve come.

Srinivasa Ramanujan's Contributions in Mathematics

 

 Ramanujan, one of the elegant Mathematician of India was born in Erode on 22nd December 1887. Erode is a small village (in that time), 400 Km away from Tamilnadu’s present capital Chennai. His father was a clerk in Kumbkonam.At the age of five, Srinivasa Ramanujan made his first appearance in school as a student. It was only a matter of time before it came to be known that he had extraordinary talent. He showed flashes of brilliance which were not to be seen in any ordinary kid at that age. He completed his primary education in a couple of years and then went to Town High School for further studies. He showed extraordinary liking for mathematics. When he was yet in school, he mathematically calculated the approximate length of earth's equator. He very clearly knew the values of the square root of two and the pie value. At the age of 16, he got scholarship. But his love only for mathematics cost him the scholarship as he neglected and failed in other subjects. His loss of scholarship was a great blow to him. He could not afford to study on his own. He had to find work and leave studies for good. He found a job of an accounts clerk in the office of the Madras Port Trust. Despite being rejected two times, his work was recognized by both G. H. Hardy and J. E. Littlewood and he went to England in 1914. In 1916,he was awarded with a degree of B.Sc. (later named Ph.D.) by Cambridge University for his work on highly composite number. In 1916, when he was at his best while working with his colleagues Hardy & Littlewood, he met with health problems. He was hospitalized in Cambridge and was diagnosed with T.B. and vitamin deficiency. After two years struggle, in 1919, he showed some recovery and he decided to return back to India. However, the improvement was temporary and after his arrival at Bombay, his health deteriorated again and finally he passed away on 26th April, 1920.His main contributions in mathematics lie in the field of Analysis, Infinite Series, Number Theory & Game Theory. His geniusness was that he discovered his own theorems. Due to his great achievements in the field of Mathematics, Indian govt. decided to celebrated his birthday 22nd December as Mathematics Day. ISTE, New Delhi and NBHM, Mumbai have taken initiative to hold Mathematical competition for students as well as teachers of colleges on the name of Srinivasa Ramanujan from 2012 to till date so that students and teachers of India know about the legacy of such great mathematician of India.

I. Hardy-Ramanujan Number

Once Hardy visited to Putney where Ramanujan was hospitalized. He visited there in a taxi cab having number 1729. Hardy was very superstitious due to his such nature when he entered into Ramanujan’s room, he quoted that he had just came in a taxi cab having number 1729 which seemed to him an unlucky number but at that time, he prayed that his perception may go wrong as he wanted that his friend would get well soon, but Ramanujan promptly replied that this was a very interesting number as it is the smallest number which can be expressed as the sum of cubes of two numbers in two different ways as given below:

1729 =1³+12³=10³+9³     

Later some theorems were established in theory of elliptic curves which involves this fascinating number.

II. Goldbach’s Conjecture   

Goldbach’s conjecture is one of the important illustrations of Ramanujan contribution towards the proof of the conjecture. The statement is every even integer >2 is the sum of two primes, that is, 6 = 3+3. 

Ramanujan and his associates had shown that every large integer could be written as the sum of at most four (Example: 43 = 2+5+17+19).

III. Theory of Equations

Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quadratic. He derived the formula to solve biquadratic equations. The following year, he tried to provide the formula for solving quintic but he couldn’t as he was not aware of the fact that quintic could not be solved by radicals.

IV. Ramanujan-Hardy Asymptotic Formula

Ramanujan’s one of the major work was in the partition of numbers. By using partition function p(𝑛), he derived a number of formulae in order to calculate the partition of numbers. In a joint paper with Hardy, Ramanujan gave an asymptotic formulas for p(𝑛). In fact, a careful analysis of the generating function for p(𝑛) leads to the Hardy-Ramanujan asymptotic formula given by

                                            p(n) ~ 1/(4n√3) e^π√(2n/3) ,  n→∞

In their proof, they discovered a new method called the ‘circle method’ which made fundamental use of the modular property of the Dedekind η-function. We see from the Hardy-Ramanujan formula that p(𝑛) has exponential growth. It had the remarkable property that it appeared to give the correct value of p(𝑛) and this was later proved by Rademacher using special functions and then Kenono gave the algebraic formula to calculate partition function for any natural number 𝑛.

 V. Ramanujan’s Congruences

Ramanujan’s congruences are some remarkable congruences for the partition function. He discovered the congruences

                        (5𝑛+4) ≡0 (𝑚𝑜𝑑 5)

                          (7𝑛+5) ≡0 (𝑚𝑜𝑑7)

                         (11𝑛+6) ≡0(𝑚𝑜𝑑 11), ∀𝑛∈𝑁.

In his 1919 paper, he gave proof for the first two congruences using the following identities using Pochhammer symbol notation. After the death of Ramanujan, in 1920, the proof of all above congruences extracted from his unpublished work.

VI. Highly Composite Numbers

A natural number n is said to be highly composite number if it has more divisors than any smaller natural number. If we denote the number of divisors of n by d(n), then we say 𝑛 є N is called a highly composite if 𝑑 (𝑚) <𝑑(𝑛) ∀𝑚 < 𝑛 where 𝑚, n∊ N. For example, 𝑛 = 36 is highly composite because it has (36) = 9 and smaller natural numbers have less number of divisors. If

                       n = 2^k₂  3^k₃…p^k_p    (by Fundamental theorem of Arithmetic)

is the prime factorization of a highly composite number 𝑛then the primes 2,3,..., 𝑝 form a chain of consecutive primes where the sequence of exponents is decreasing; i.e.𝑘₂ ≥ 𝑘₃ ≥.....≥ k_p and the final exponent 𝑘_p is 1, except for 𝑛 = 4 and 𝑛 = 36.

VII. Some Other Contributions

Apart from the contributions mentioned above, he worked in some other areas of mathematics such as hypogeometric series, Bernoulli numbers, Fermat’s last theorem. He focused mainly on developing the relationship between partial sums and products of hyper-geometric series. He independently discovered Bernoulli numbers and using these numbers, he formulated the value of Euler’s constant up to 15 decimal places. He nearly verified Fermat’s last theorem which states that no three natural number 𝑥, 𝑦 and 𝑧 satisfy the equation 𝑥ⁿ +𝑦ⁿ  = 𝑧ⁿ for any integer 𝑛 > 2.

PUZZLE

 

Top 10 Mathematical Innovations

 

1. Arabic numerals

 Great advances in Western European science followed the introduction of Arabic numerals by the Italian mathematician Fibonacci in the early 13th century. He learned them from conducting business in Africa and the Middle East. Of course, they should really be called Hindu numerals because the Arabs got them from the Hindus. In any case, mathematics would be stuck in the dark ages without such versatile numerals.

2. Calculus (Isaac Newton, Gottfried Leibniz)

Newton gets all the credit, even though Leibniz invented calculus at about the same time, and with more convenient notation (still used today). In any event, calculus made all sorts of science possible that couldn’t have happened without its calculational powers. Today everything from architecture and astronomy to neuroscience and thermodynamics depends on calculus.

4. Zero and 3. Negative numbers (Brahmagupta)

Brahmagupta, a seventh-century Hindu astronomer, was not the first to discuss negative numbers, but he was the first to make sense of them. It’s not a coincidence that he also had to figure out the concept of zero to make negative numbers make sense. Zero was not just nothingness, but a meaningful number, the number you get by subtracting a number from itself. Zero was not just a placeholder, writes Joseph Mazur in his new book Enlightening Symbols. For what may have been the first time ever, there was a number to represent nothing.

5. Decimal fractions (Simon Stevin, Abu’l Hasan Al-Uqlidisi)

Stevin introduced the idea of decimal fractions to a European audience in a pamphlet published in 1585, promising to teach how all Computations that are met in Business may be performed by Integers alone without the aid of Fractions. He thought his decimal fraction approach would be of value not only to merchants but also to astrologers, surveyors and measures of tapestry. But long before Stevin, the basic idea of decimals had been applied in limited contexts. In the mid-10th century, al-Uqlidisi, in Damascus, wrote a treatise on Arabic (Hindu) numerals in which he dealt with decimal fractions, although historians differ on whether he understood them thoroughly or not.

6. Binary logic (George Boole)

Boole was interested in developing a mathematical representation of the “laws of thought,” which led to using symbols (such as x) to stand for concepts (such as Irish mathematicians). He hit a snag when he realized that his system required x times x to be equal to x. That requirement pretty much rules out most of mathematics, but Boole noticed that x squared does equal x for two numbers: 0 and 1. In 1854 he wrote a whole book based on doing logic with 0s and 1s — a book that was well-known to the founders of modern computer languages.

7. Non-Euclidean geometry (Carl Gauss, Nikolai Lobachevsky, János Bolyai, Bernhard Riemann)

Gauss, in the early 19th century, was probably the first to figure out an alternative to Euclid’s traditional geometry, but Gauss was a perfectionist, and perfection is the enemy of publication. So Lobachevsky and Bolyai get the credit for originating one non-Euclidean approach to space, while Riemann, much later, produced the non-Euclidean geometry that was most helpful for Einstein in articulating general relativity. The best thing about non-Euclidean geometry was that it demolished the dumb idea that some knowledge is known to be true a priori, without any need to check it out by real-world observations and experiments. Immanuel Kant thought Euclidean space was the exemplar of a priori knowledge. But not only is it not a priori, it’s not even right.

8. Complex numbers (Girolamo Cardano, Rafael Bombelli)

Before Cardano, square roots of negative numbers had shown up in various equations, but nobody took them very seriously, regarding them as meaningless. Cardano played around with them, but it was Bombelli in the mid-16th century who worked out the details of calculating with complex numbers, which combine ordinary numbers with roots of negative numbers. A century later John Wallis made the first serious case that the square roots of negative numbers were actually physically meaningful.

9. Matrix algebra (Arthur Cayley)

An ancient Chinese math text included matrix-like calculations, but their modern form was established in the mid-19th century by Cayley. (Several others, including Jacques Binet, had explored aspects of matrix multiplication before then.) Besides their many other applications, matrices became extremely useful for quantum mechanics. In fact, in 1925 Werner Heisenberg reinvented a system identical to matrix multiplication to do quantum calculations without even knowing that matrix algebra already existed.

10. Logarithms (John Napier, Joost Bürgi, Henry Briggs)

A great aid to anybody who multiplied or messed with powers and roots, logarithms made slide rules possible and clarified all sorts of mathematical relationships in various fields. Napier and Bürgi both had the basic idea in the late 16th century, but both spent a couple of decades calculating log tables before publishing them. Napier’s came first, in 1614. Briggs made them popular, though, by recasting Napier’s version into something closer to the modern base-10 form.