# Mathematics II PEAK 2017 – 2018 (1)

## Atsushi Matsuo

Continue to Mathematics II PEAK 2017 – 2018 (2)
Previous Year: Mathematics II PEAK 2016 – 2017 (1)
Next Year: Mathematics II PEAK 2018 – 2019 (1)
NOTICES   CLASS SCHEDULE   COURSE INFORMATION   LECTURE PLAN   TUTORIAL   PAST NOTICES
• NOTICES

• CLASS SCHEDULE

The schedule may change.

1. 2017/09/27 (Wed)  Guidance. Sets and maps.
2. 2017/10/04 (Wed)  Sets and maps, continued.
3. 2017/10/11 (Wed)  Linear transformations of the coordinate plane. Minitest 1.
4. 2017/10/18 (Wed)  Geometry of the coordinate space. Minitest 2.
5. 2017/10/25 (Wed)  Geometry of the coordinate space, continued. Matrices and linear maps.
6. 2017/11/01 (Wed)  Matrices and linear maps, continued. Minitest 3.
7. 2017/11/08 (Wed)  Elementary operations on matrices. Minitest 4.
8. 2017/11/22 (Wed)  Inverse matrices. Minitest 5.
9. 2017/11/29 (Wed)  Inverse matrices, Continued. Bases of vector spaces. Minitest 6.
10. 2017/12/06 (Wed)  Bases of vector spaces, Continued.
11. 2017/12/13 (Wed)  Application of bases.
12. 2017/12/20 (Wed)  Orthogonal bases.
13. 2018/01/10 (Wed)  Determinants and volumes.
14. 2018/01/17 (Wed)  Final Exam.
Time and Place: 08:40—10:10  Room 157
15. 2018/04  Make-up Exam.

• 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 as shown below.

Dates:  Open on the dates marked by pink in the tables below.
 (The room is closed on the cyan dates.)
Time:  5th period for Mondays and Wednesdays, and 11:15-13:00 for Fridays.
Place:  KIBER 109  (Room 109 of ¶ð¾ì¹ñºÝ¶µ°é¸¦µæÅï)

September
MonTueWedThuFriSatSun
123
45678910
11121314141617
18192021222324
252627282930

October
MonTueWedThuFriSatSun
1
2345678
9101112131415
16171819202122
23242526272829
3031

November
MonTueWedThuFriSatSun
12345
6789101112
13141516171819
20212223242526
27282930

December
MonTueWedThuFriSatSun
123
45678910
11121314151617
18192021222324
25262728293031

January
MonTueWedThuFriSatSun
1234567
891011121314
15161718192021
22232425262728
293031

• PAST NOTICES
• 2017/10/12   Correction.
• Lecture 2, Questions and Answers.
• Question 2.2, Answer  (1)  2[1,0]T + 3[0,1]T   (4)  -[2,1]T + 2[1,1]T
• Question 2.5, Answer  (1) ↔ (2)
I thank the students who pointed out the errors.
• 2017/10/18   Correction.
• Lecture 3, Summary.
• Question 3.4, 2nd line: ‘a hyperplane’ should be ‘a line.’
• Lecture 3, Questions and Answers.
• Question 3.3, Answer, 1st line: remove ‘=[2,2,1]T
• Question 3.4, 2nd line: ‘a hyperplane’ should be ‘a line.’
• Question 3.4, Answer, 3rd line: ‘+t[1,1]T=[t,t+1]T’should be ‘+t[2,1]T=[2,t+1]T
I thank the students who pointed out the errors.
• 2017/11/08
• No class on November 15th, when supplementary classes will take place.
• 2017/11/30   Correction.
• Lecture 6, Questions and Answers.
• Question 6.1, Answer (1), 1st line: ‘injective’ should be ‘surjective.’
• Question 6.1, Answer (1), 1st line: ‘[y, y-x]T’ should be ‘[y-x, x]T.’
• Question 6.1, Answer (2), 2nd line: ‘twice’ should be ‘twice the first entry.’
• Question 6.2, Answer (5): ‘rank A = 2, No, No’ should be ‘rank A = 3, Yes, No.’
I thank the student who pointed out errors.
• 2018/ 1/20   On marking the exam.
• Problem 1 (1): The cross product is [2,-1,-2]T, and the unit vectors are [2/3,-1/3,-2/3]T, and [-2/3,1/3,2/3]T,.
One can partially verify his results by calculating the inner product with the given vectors, although it does not totally guarantee the correctness of the cross product.

• Problem 1 (2): The minimal polynomial is t3 as E, A, A2 are linearly independent and A3=O. The powers of A are given by

A0  =  1 0 0 0 1 0 0 0 1
,    A1  =  1 1 -1 1 0 -1 1 1 -1
,    A2  =  1 0 -1 0 0 0 1 0 -1
,    An  =  0 0 0 0 0 0 0 0 0
,  (n ≥ 3).
• Problem 2 (1): The angle is 3π/4, i.e., 135°.
Some students errorniously considered the absolute value of what is to be equal to cosine of the angle.

• Problem 2 (2): The orthogonal vectors obtained by Gram–Schmidt process are [1,2,1]T, [1,-1,1]T, [1,0,-1]T.
One can partially verify his results by calculating the inner products of the resulting vectors with each other, although it does not totally guarantee the correctness of the results of Gram–Schmidt process.

• Problem 3 (1): The inverse is given by

A-1 =  1 3 2 3 3 1 1 2 1
One can verify his result by calculating the product with the given matrix.
• Problem 3 (2): rank A=3 and dim Ker AT=1.

• Problem 4 (1): The row reduced form is given by  1 0 1 1 0 1 1 -1 0 0 0 0 0 0 0 0

• Problem 4 (2): A basis of Im A is given by [1,2,3,2]T, [0,1,1,2]T.
As the first and the second columns of the row reduced form give a basis of the image of the row reduced form, the first and the second columns of A gives a basis of the image of A. Some students errorniously answered the first and the second columns of the row reduced form instead of those of A.

• Problem 4 (3): A basis of Ker A is given by [-1,-1,1,0]T, [-1,1,0,1]T.
Some students used properties of the dimension instead of the definition of linear dependence or spanning. Such a solution does not fulfill the requiremen of the problem.

• Problem 5: No one received a positive point.