# Mathematics II PEAK 2015 – 2016 (1)

## Atsushi Matsuo

Continue to Mathematics II PEAK 2015 – 2016 (2)
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NOTICES   CLASS SCHEDULE   COURSE INFORMATION   LECTURE PLAN   TUTORIAL   PAST NOTICES
• NOTICES

• 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.
• CLASS SCHEDULE

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
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

Exam-period Exam Timetable (include Exam room)
• COURSE INFORMATION

• 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.
• LECTURE PLAN

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

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 ¶ð¾ì¹ñºÝ¶µ°é¸¦µæÅï

The scheduled days are marked by pink:
October
MonTueWedThuFriSatSun
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262728293031

November
MonTueWedThuFriSatSun
1
2345678
9101112131415
16171819202122
23242526272829
30

December
MonTueWedThuFriSatSun
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28293031
• PAST NOTICES
• 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=en have solutions p1,...,pn, 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 ATPT=E. Now the argument 2 implies that AT 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 xRn

• December 3, 2015   Correction