Mathematics II PEAK 2015 – 2016 (2)

• Sepember 23, 2016  
  Make-up Exam for S semester is scheduled as follows:

  Date: 2016/10/06 (Thu) 
  Time: 10:35-12:05 

The schedule may change.

1. 2016/04/11 (Mon)  Lecture 10. Determinants and volumes (review)
2. 2016/04/18 (Mon)  Lecture 10. Determinants and volumes (continued)
3. 2016/04/25 (Mon)  Lecture 11. Complex numbers
   
4. 2016/05/02 (Mon)  Lecture 12. Criteria of regularity
   
5. 2016/05/09 (Mon)  Lecture 13. Eigenvalues and eigenvectors.
   
6. 2016/05/23 (Mon)  Lecture 14. Diagonalization of square matrices.
   
7. 2016/05/30 (Mon)  Exercise Session 3. Eigenvalues and diagonalization of matrices.
   
8. 2016/06/06 (Mon)  Lecture 15. Quadratic forms
   
9. 2016/06/13 (Mon)  Lecture 16. Genralized eigenspace decomposition
   
10. 2016/06/20 (Mon)  Lecture 17. Jordan canonical form
    
11. 2016/06/27 (Mon)  Lecture 18. Exponential of square matrices.
12. 2016/07/04 (Mon)  Lecture 19. Linear differential equations with constant coefficients.
13. 2016/07/11 (Mon)  Exercise Session 4
    
14. 2016/07/25 (Mon)  Exam for S semester 
    Time and Place: 13:10-14:40  in Room 1211 
15. 2016/10/06 (Thu)  Make-up Exam for S semester 
    Time and Place: 10:25-11:55  10:35-12:05  in Room *** 

• Continutation of introductory course in Linear Algebra.
• Room 512, 13:00 - 14:45, Monday.
• Students will study the properties of vectors, matrices and determinants as well as the concepts of abstract vector spaces and linear maps on such spaces. Various applications of these concepts will also be presented.
• The lecture will be in traditional lecture style using the blackboard.
• The course will not be based on a particular textbook. Recommended references will be presented in the class.
• Evaluation will be based on written examination at the end of the semester.

The following topics will be covered by the course.

1. Complex numbers and fundamental theorem of algebra
2. Criteria of regularity
3. Eigenvalues and eigenvectors
4. Diagonalization of square matrices
5. Quadratic forms
6. Generalized eigenspace decomposition
7. Jordan canonical form
8. Applications of canonical forms

Tutorial by assistant professors will take place in the following schedule:

Days: Every Monday, Wednesday, and Friday during the semester.

Time: 5th period

Place: Room 109 of 駒場国際教育研究棟

The scheduled days are marked by pink:
April

Mon	Tue	Wed	Thu	Fri	Sat	Sun
				1	2	3
4	5	6	7	8	9	10
11	12	13	14	15	16	17
18	19	20	21	22	23	24
25	26	27	28	29	30

May

Mon	Tue	Wed	Thu	Fri	Sat	Sun
						1
2	3	4	5	6	7	8
9	10	11	12	13	14	15
16	17	18	19	20	21	22
23	24	25	26	27	28	29
30	31

June

Mon	Tue	Wed	Thu	Fri	Sat	Sun
		1	2	3	4	5
6	7	8	9	10	11	12
13	14	15	16	17	18	19
20	21	22	23	24	25	26
27	28	29	30

July

Mon	Tue	Wed	Thu	Fri	Sat	Sun
				1	2	3
4	5	6	7	8	9	10
11	12	13	14	15	16	17
18	19	20	21	22	23	24
25	26	27	28	29	30	31

• April 12, 2016  
  Make-up Exam for the last semester
  Date: Wednesday, April 27th
  Time: 10:25 – 11:55
  Place: to be announced

• April 12, 2016   (A comment added on April 13, Time changed on April 14)
  Extra session for the students who failed the last exam:
  Date: Thursday, April 14th
  Time: 16:50 – 18:35
  Place: Room 115
  In this extra session, I will answer to questions on the homeworks in the last semester. The students are required to have done the homeworks in advance.
• May 24, 2016  
  The morning of May 30, 2016 is `Supplementary Classes' day, but the afternoon is NOT.
  We will have the exercise session on May 30.
• June 6, 2016   Correction.
  On the blackboard, ‘bilinear form’ should have been ‘symmetric bilinear form’.
• June 27, 2016   Correction.
  On the blackboard, ‘triangular inequality’ should have been ‘triangle inequality’.
• July 2, 2016   Correction.
  Question 18.2 should be: ‘Find the exponential exp tA for each matrix A.’
• July 6, 2016   Correction.
  Question 13.3 (2): The answer should read ‘(t-4)t³. t=4,0’ instead of ‘(t-3)(t+1)³. t=4,0’
• July 15, 2016   Correction.
  Question 19.2 (2): ‘-2’ of RHS should read ‘-4’.
• July 15, 2016   Answers for Lecture 19.
• July 15, 2016   Office Hour.
  Time: July 18, 3rd period, Place: 21 KOMCEE West 3F K303
• July 18, 2016   Correction.
  Question 18.1 (5): ‘z(t) = [ (x + yt) e^t, ye^t ]^T’ should read ‘z(t) = [ (x + (x+y)t) e^t, (y - (x+y)t) e^t ]^T’.
• July 27, 2016   On marking the Exam. (Revised, July 28)
  1. By calculating the characteristic polynomial, one see that the eigenvalues are 3 and −3 with multiplicities 2 and 1, respectively. So the signature is (2,1,0). For (2), one needs to find an orthogonal basis for the 2-dimensional eigenspace belonging to the eigenvalue 3, say, by Gram-Schmidt process. Then normalize the vectors to obtain an orthonormal basis.
  2. Many students seem to have wasted time by calculating the characteristic polynomial. In fact, for the given matrix, calculating the minimal polynomial directly is much easier than calculating the characteristic polynomial by using determinant.
  3. Unfortunately, the matrix of Problem 3 had different interpretations: some students interpreted the value i in the (i,j)-th entry for i=j as the imaginary unit, and others as the i of (i,j). I treated and marked fair for the both interpretations.
  4. The problem is on a linear transformation of an infinite-dimensional vector space. One may use matrices by restricting the space to the finite-dimensional subspace of polynomials of degree less than n for every n, but one needs to argue more than manipulating n by n matrices in order to obtain the results for the whole space.
  5. First note that there are matrices A and B with the same characteristic polynomial but not similar to each other. It is indeed crucial to use the condition q ≠ 0. So any solution without a use of this condition is wrong. Now, it follows from the conditions that the real matrices A and B have the same pair of distinct nonreal eigenvalues. So A and B are both diagonalizable by complex nonsingular matrices, and hence the matrices are similar by a complex nonsingular matrix. Once this has been shown, the final task is to show that A and B are actually similar by a real nonsingular matrix. One can also find a real nonsingular matrix giving the similarity without addressing to complex numbers at all.