Points are reduced in such cases even though the final result were correct.
For fairness, points are equaly reduced if only the final result is written without mentioning the formula or any other reason justying the result.
Problem 1 (3)
For instance,
[x, y, z]T = [1, 0, 0]T + t1 [1, -1, 0]T + t2 [1, 0, -1]T.
Problem 1 (4)
For instance, (3n), (n 3n-1).
The pair (3n), (n 3n) also gives a right answer.
Problem 2 (1)
A1000
= A3·333+1
= (A3)333A
= (-1)333A
= -A =
|
| . |
One may also calculate it as A1000 = A6·166+4 = A4.
Problem 2 (2)
Problem 3 (1)
Rank A = 2.
Problem 3 (2)
For instance,
[-2, 1, 0, 0, 0]T, [-3, 0, 1, 1, 0]T, [1, 0, -2, 0, 1]T.
Problem 4 (1)
The minimal polynomial, say in t, is t2 - 2 t + 1.
One may answer it in the factored form as (t - 1)2.
Choice of indeterminate does not matter, as it is not specified in the question, so, for instance, the answer may be (x - 1)2, (λ - 1)2, etc.
Some students tried to compute the characteristic polynomial instead of the minimal polynomial. The two polynomials are related, but need not agree, and they indeed disagree for the matrix given in the question.
Problem 4 (2)
Omitted.
The answer must be based on the definition of linear independence.
Problem 5 (1)
dim Im F=2, dim Ker F=1. Proof of F being linear omitted.
In the context of the problem, the sum of two polynomials must be of the form p(x) + q(x) for polynomials p(x) and q(x) both in the same indeterminate x. However, some students erroneously considered sums such as p(x) + q(y).
Problem 5 (2)
Omitted.
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- CLASS SCHEDULE
The schedule may change.
-
2019/09/25 (Wed) Guidance, Sets and maps
-
2019/10/02 (Wed) Linear transformations of coordinate plane
-
2019/10/09 (Wed) Geometry of coordinate spaces
-
2019/10/16 (Wed) Matrices and linear maps
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2019/10/23 (Wed) Linear subspaces of coordinate spaces
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2019/10/30 (Wed) Row reduction of matrices
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2019/11/06 (Wed) Inverse matrices
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2019/11/20 (Wed) Bases and dimension
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2019/11/27 (Wed) Bases and dimension (continued)
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2019/12/04 (Wed) Rank and nullity
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2019/12/11 (Wed) Trial exam; Linear recurrence relations
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2019/12/18 (Wed) Linear recurrence relations (continued)
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2020/01/08 (Wed) Powers of square matrices
-
2020/01/22 (Wed) Final Exam.
13:10กม14:40 at room 531
-
2020/05/26 (Tue) Make-up Exam.
08:40กม10:10 online
Academic Calender 2019—2020
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- COURSE INFORMATION
- 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:
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- PAST NOTICES
- Students are supposed to participate in Mathematics Exercise Session scheduled as follows:
- Thursdays, Period 5 (16:50-18:35)
- Starting from September 26, 2019 and running until January 9, 2020
- Room 109, KIBER (Komaba International Building for Education and Research)
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