Mathematics II PEAK 2018 – 2019 (1)

• 2019/01/31   On marking the make-up exam:
  • Problem 1 (1) The cross product is given by [5,-2,-1]^T.
  • Problem 1 (2) The equation is 3x-2y+z=1. The plane is NOT a linear subspace.
  • Problem 2 (1) The matrix representation is:

    -1/2	-(√3)/2
    -(√3)/2	1/2

  • Problem 2 (2) The signature is (1,1,1) (or (1,1)).
  • Problem 3 (1) A⁴ is

    16	0	0
    0	16	0
    0	0	16

    A^-1 is

    1/2	1/2	0
    -1/4	-1/4	1/2
    1/8	-3/8	1/4

  • Problem 4 (1) The row reduced form is

    1	0	-1	-1
    0	1	-1	2
    0	0	0	0
    0	0	0	0

  • Problem 4 (2) For instance, [1,1,1,0]^T,[1,-2,0,1]^T is such a basis of Ker A.
  • Problem 4 (3) For instance, [1,1,0,2]^T,[2,0,1,1]^T is such a basis of Im A.
    Note: [1,0,0,0]^T,[0,1,0,0]^T is not a basis of Im A, for the vectors do not belong to Im A.
  • According to the notices described on the cover, points are reduced if the logical symbol ⇒ is used in an improper way.
• 2019/01/31   On marking the exam:
  • Problem 1 (1) The angle is π/4   i.e., 45°.
  • Problem 1 (2) The area is 7.
  • Problem 1 (3) The matrix is given by
    

    -(√3)/2	-1/2
    1/2	-(√3)/2

  • Problem 1 (4) The dimension is 3.
  • Problem 2 (1) For the given matrix A, one has A¹⁰ = (A³)³A = 2³A = 8A =
    

    8	8	-8
    8	-16	24
    8	-8	8

  • Problem 2 (2) A^-1 is given by
    

    1	2	2
    2	2	1
    2	3	2

  • Problem 3 (1) The representation matrix is given by
    

    1	-2	1
    -2	1	1
    1	1	-2

    and hence a basis of the radical is given for instance by [1,1,1]^T.
  • Problem 3 (2) The minimal polynomial is given by t²-2t+1.
• 2018/09/24
  • Students are supposed to
    — take notes from the blackboard during class.
    — bring their own notes in asking questions on the contents of the lecture.
    — attend the exercise session and tutorial in 5th period.

The schedule may change.

1. 2018/09/27 (Thu)  Sets and maps
2. 2018/10/04 (Thu)  Linear transformation of coordinate plane
3. 2018/10/11 (Thu)  Geometry of coordinate spaces
4. 2018/10/18 (Thu)  Matrices and linear maps
5. 2018/10/25 (Thu)  Linear subspaces of coordinate spaces
6. 2018/11/01 (Thu)  Elementary operations on matrices
   2018/11/08 (Thu)  (No class)
7. 2018/11/15 (Thu)  Bases and dimension
   2018/11/22 (Thu)  (No class)
8. 2018/11/29 (Thu)  Rank of matrices
9. 2018/12/06 (Thu)  Signature of quadratic forms
10. 2018/12/13 (Thu)  Recursions and differential equations
11. 2018/12/20 (Thu)  Direct sum decomposition
12. 2018/12/27 (Thu)  Differential equations & Trial Exam
13. 2019/01/10 (Thu)  Minimal polynomials & Exercise Session
14. 2019/01/17 (***)  Final Exam.
    Time and Place: 08:40—10:10  Room 514 
15. 2019/04/** (***)  Make-up Exam.

• Reference Books
  In case you wish to look at related books, I pick up the following although they do not match the contents of this course:
  • S. Lipschutz and M. Lipson, Schaum's Outline of Linear algebra, McGraw-Hill.
  • G. Strang, Linear Algebra and Its Applications, 5th edition,Wellesley-Cambridge Press.

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