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Books for TRB syllabus Topic Authors Analysis 1. Rudin 2. Apostol Algebra 1. Herstein 2. Hoffman Kunze Complex Analysis Ahlfors Topology Munkres Functional Analysis Simmons Differential Equations 1. Sneddon 2. coddington 3. Sankara rao 4. Simmons 5. Grewal (Enggineering Mathematics) Differential Geometry 1. Willmore 2. Somasundaram Probability and Statistics 1. Hogg 2. Gupta Kapoor 3. Grewal (Enggineering Mathematics) Linear programming 1.Sharma, J.K 2. Taha 3. Grewal (Enggineering Mathematics) Numerical Analysis 1. Brian Bradie 2. Jain and Iyyenger 3. Grewal (Enggineering Mathematics) Graph Theory Bondy and Murty Mechanics related subjects 1. Goldstein 2. ...

Let T  be a nilpotent linear operator on the vector space $\mathbb{ R}^5$  (i.e., $ T^k=0$  for some k). Let $d_i$  denote the dimension of the kernel of $ T^i$ . Which of the following can possibly occur as a value of $ (d_1,d_2,d_3)$ ?  1. (1,2,3) 2. (2,3,5) , 3. (2,2,4) , 4. (2,4,5) For the excellent proof using Jordan Canonical form visit  https://nbhmcsirgate.theindianmathematician.com/2020/04/nbhm-2020-part-c-question-26-solution.html#.XqCM3QR0R10.whatsapp Here we present a proof (incomplete in preciousness) on the fact "A linear transformation is completely determined by its behaviour on a basis."

Many of the students are asking their doubts by sending a picture in whatsapp. Here I post the Hints for the answers, so that others also can use the problems. As I have too little time to type all in Latex, here simply I post the photo.  Any more suggestion, comments are welcome. Also I post Problems and Hints, that I found in some groups asked and solved by others. On behalf of myself and readers of this post, I express my sincere thanks to them.  *Problems will be updated regularly* Warning: ALL THE ANSWERS AND HINTS NEED  NOT BE CORRECT. VERIFY THOROUGHLY.  ALSO IF YOU FIND ANY MISTAKES PLEASE WARN ME. I TRY TO CORRECT.  Hint:  https://math.stackexchange.com/questions/697738/dimension-of-the-vector-space-of-homogeneous-polynomials *End of Problem* Hint: Use M-l inequality in complex integration. Also Here the question is incorrect. Instead of |z|=1, take |z|=2. https://math.stackexchange.com/questions/666229/what-is-ml-inequalit...

A CSIR question... Let A be a 2x2 matrix over real satisfies det(A+I)=1+det(A), then 1. det A = 0 2. A= 0 3. tr A = 0 4. A is nonsingular Basic knowledge on eigen values will give you the answer. BUT Is it possible to find the answer, by the method of elimination? Yes... This is a part B question, in CSIR examination... (Only one ans is correct). Suppose option 2 is correct, then 1 and 3 are also correct, this eliminates option 2. Take A as 0, this eliminates option 4. So the ans is either 1 or 3... Now we try to produce example that satisfies any one but not other. If that example satisfies hypothesis of the question, then that should be the answer. Can you produce such example??? _________________________________________________________________________________ In CSIR 2018 Dec Mathematics, it was asked to find the number of homomorphisms from A5 to S4. Immediately our mind thinks about any formula to find the ans, as we have in Zn type groups. B...

Let $F$ be a field. Define Splitting field - Finite extension - Algebraic extension - Simple extension - Normal extension - Separable extension - Galois group - Solvable group 1. Differentiate Algebraic integer and Algebraic number. Justify the following Statements: 1. The splitting field of a polynomial over $F$ need not be a finite extension. [TRB] 2. An algebraic extension is always a finite extension. [TRB] 3. $\mathbb{R}$ is an algebraic extension of $\mathbb{Q}$. 4. Any finite extension of a field is a simple extension. [TRB] 5. Any finite extension of a field of characteristic zero is a simple extension. [TRB] 6. Normal extension of a field is always a finite extension. [TRB] 7. Every algebraic integer is algebraic number. 8.    Every  group with order less than 60 is solvable. 9. $S_n$ is solvable for $n$= (4/6/60/30) 10. Which of the following group os not solvable? ($S_3/S_5/Z_4/Quaternion group $)