Mathematics II PEAK 2015 – 2016 (1)

• 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.

1. 2015/09/14 (Mon)  Lecture 1.
   Binary operations and vector spaces
2. 2015/09/21 (Mon)  Lecture 1 & 2.
   Matrices and linear maps
    > Lecture 1 summary, Lecture 1 supplements, Lecture 2 summary
3. 2015/09/28 (Mon)  Lecture 2 & 3.
   Systems of linear equations and matrices
    > Lecture 1 answers, Lecture 2 revised summary, Lecture 2 answers, Lecture 3 summary
4. 2015/10/05 (Mon)  Lecture 3 & 4.
   Elementary operations on matrices
    > Lecture 4 summary
5. 2015/10/12 (Mon)  Lecture 4
    > Lecture 3 answers, Lecture 4 revised summary
6. 2015/10/19 (Mon)  Exercise Session 1
    > Exercise session 1 problems
7. 2015/10/26 (Mon)  Lecture 5.
   Bases and dimensions
    > Lecture 4 answers, Exercise session 1 answers, Lecture 5 summary
8. 2015/11/02 (Mon)  Lecture 6.
   Change of bases
    > Lecture 5 answers, Lecture 6 summary
9. 2015/11/09 (Mon)  Lecture 7.
   Ranks of matrices
    > Lecture 6 answers, Lecture 7 summary
10. 2015/11/16 (Mon)  Lecture 8.
    Orthonormal bases
     > Lecture 7 answers, Lecture 8 summary
11. 2015/11/30 (Mon)  Lecture 8 & 9.
    Vector product of spatial vectors, Lecture 8 answers, Lecture 9 summary
12. 2015/12/07 (Mon)  Exercise Session 2
     > Exercise session 2 problems, Lecture 9 answers
    
13. 2015/12/14 (Mon)  Lecture 10.
    Determinants and volumes
     > Exercise session 2 solutions, Lecture 10 summary, Lecture 10 answers
    
14. 2015/12/21 (Mon)  Exam for A semester 
    Time and Place: 13:00-14:30  in Room 102 
15. 2016/ 4/27 (Wed)  Make-up Exam for A semester 
    Time: 10:25-11:55 

• 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. Binary operations and vector spaces
2. Matrices and linear maps
3. Systems of linear equations and matrices
4. Elementary operations on matrices
5. Bases and dimensions
6. Change of bases
7. Ranks of matrices
8. Orthonormal bases
9. Vector product of spatial vectors
10. Determinants and volumes

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

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

Time: 5th period

Place: Room 109 of 駒場国際教育研究棟

The scheduled days are marked by pink:
October

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

November

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

December

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

• September 18, 2015

  • Correction to Lecture 2 summary: 4 (1) associative law (CB)A=C(AB) should read (CB)A=C(BA).

• September 21, 2015

  • Two errors in Supplements to Lecture 1 were corrected during the class.
  • Answers to questions will be given next week.

• October 18, 2015

  • Errors in the revised summary to Lecture 4 were corrected during the class on October 12.
• October 19, 2015   On the Tutorial
  Tutorial by assistant professors will take place in the following schedule: (omitted)

• November 9, 2015   Correction   (modified on November 10)
  There was a serious mistake in the argument for the equivalence at the very last part of the lecture: I implicitly used the invertibility of the matrix P, which was not assumed in the argument. One of the correct arguments is as follows:

  1. Assume A has full rank. Then by Gaussian elimination PA=E for some invertible matrix P. Since P is invertible, we have A=P^-1 and hence AP=E. (Thus PA=E=AP, so A is invertible and A^-1=P.)
  2. Assume AP=E for some matrix P. Then the systems Ax=e₁,...,Ax=e_n have solutions p₁,...,p_n, respectively. So the system Ax=b has a solution for any b. This implies that A has full rank.
  3. Assume PA=E for some matrix P. Exchanging A and P in the argument 2, we see that P has full rank. So P is invertible and P^-1=A by the argument 1. In particular, we have AP=E and hence A has full rank by the argument 2 again.
  Therefore, the following conditions are equivalent to each other:
  • A has full rank.
  • A is invertible.
  • AP=E for some matrix P
  • PA=E for some matrix P
  One may consider the transpose of the argument 2 as an alternative of the argument 3: Assume PA=E for some matrix P. Then we have A^TP^T=E. Now the argument 2 implies that A^T has full rank. Hence A has full rank.

• November 30, 2015   Correction  
  Lecture 6, Summary, 1 (3), the last line:
  “ψ(f (v))=Aφ(v) for all v ∈ V” should read “f (φ(x))=ψ(Ax) for all x∈ Rⁿ”

• December 3, 2015   Correction  
  Lecture 8, Answer.
  • Questions 8.2, (4), third line, before the inequality: substitute “3x₂²/2”
  • Questions 8.3, (2), after the equal sign: substitute “3/5”
  • Questions 8.5, the second line of (1), before the big left bracket: insert “=(1,0,0)^T-”
  • Questions 8.5, the first and the second lines of (2), the first column vector: substitute “=(1,1,0)^T”