Norwegian University of Science and Technology
Department of Mathematical Sciences
Page 1 of 3
Contact during the exam: Mats Ehrnstrm, 735 917 44
Exam in TMA4145 Linear Methods
Wednesday, December 5, 2012
Time: 09:00 13:00
Examination aids: Code D
Grades:
Norwegian University of Science and Technology
Department of Mathematical Sciences
Page 1 of 10
Suggested Solutions
TMA4145, December 2012
Problem 1
(Overview)
For each of the following, indicate whether it is true or false (no proof required).
(i) Any ve
Norwegian University of Science and Technology
Department of Mathematical Sciences
Page 1 of 3
Contact during the exam: Mats Ehrnstrm, 735 917 44
Exam in TMA4145 Linear Methods
Friday, August 16, 2013
Time: 09:00 13:00
Examination aids: Code D
Grades: Fri
TMA4145 Linear
Methods
Autumn 2012
Norwegian University of Science
and Technology
Department of Mathematics
Exercise set 10
1 Let A Mnn (R). Use Picard iteration for x = Ax, x(0) = x0 , to nd the solution
of this initial-value problem.
2 (Friedberg 5.1:8,
TMA4145 Linear
Methods
Autumn 2012
Norwegian University of Science
and Technology
Department of Mathematics
Exercise set 13
1 (Problem 4, 2000 )
a) Find the QR-factorization of
1 1 1
A = 1 2 4 .
1 3 9
b) Which point in spancfw_(1, 1, 1), (1, 2, 3) C3 is c
TMA4145 Linear
Methods
Autumn 2012
Norwegian University of Science
and Technology
Department of Mathematics
Exercise set 7
This weeks exercises are all from old exams (its good practice!). The dates refer to which
exams they were taken from.
1 (Exercise 1
TMA4145 Linear
Methods
Autumn 2012
Norwegian University of Science
and Technology
Department of Mathematics
Exercise set 2
1 (Kreyzig, exercise 1.1.2) Does d(x, y) = (x y)2 dene a metric on the real line?
Why/why not?
2 (Kreyzig, exercise 1.1.10) Let X be
TMA4145 Linear
Methods
Autumn 2012
Norwegian University of Science
and Technology
Department of Mathematics
Exercise set 4
1 (Friedberg 1.2.18) Consider the set R2 and dene addition and scalar multiplication
on it as follows.
(x1 , x2 ) + (y1 , y2 ) = (x1
TMA4145 Linear
Methods
Autumn 2012
Norwegian University of Science
and Technology
Department of Mathematics
Exercise set 5
1 (Friedberg 1.3.8) Determine which of the following sets are subspaces of R3 under the
operations of addition and scalar multiplica
TMA4145 Linear
Methods
Autumn 2012
Norwegian University of Science
and Technology
Department of Mathematics
Exercise set 6
1 (Strang 2.3.33) Find a basis for each of these subspaces of M33 (R):
a) All diagonal matrices (aij = 0 for i = j).
b) All symmetri
TMA4145 Linear
Methods
Autumn 2012
Norwegian University of Science
and Technology
Department of Mathematics
Exercise set 3
1 (Strang 2.1.2) Which of the following subsets of R3 are vector spaces when we dene
addition and scalar multiplication as on R3 ?
G
TMA4145 Linear
Methods
Autumn 2012
Norwegian University of Science
and Technology
Department of Mathematics
Exercise set 1
1 A deck of cards can be considered a set, where each element (i.e. card) has a value
and a suit. Given an element x, xv is its valu