Linear Algebra Problems
MA2030
589
UNIVERSITY OF MORA TUW A
Faculty of Engineering Department of Mathematics B. Sc. Engineering Level 2 - Semester 2 Examination: MA 2030 LINEAR ALGEBRA Time Allowed: 2 hours
2010 September 2010
ADDITIONAL
MATERIAL: None
INSTRUCTIONS
TO CANDIDATES:
This paper contains 6 questions and 5 pages.
Answer FIVE questions and NO MORE. This is a closed book examination.
Only the calculators approved and labeled by the Faculty of Engineering are permitted. This examination accounts for 70% of the module assessment.
Assume reasonable values for any data not given in or with the examination paper. Clearly state such assumptions made on the script.
If you have any doubt as to the interpretation of the wording of a question, make your own decision, but clearly state it on the script.
- 1-
MA2030
1. a) Let a be an object and define V = {a }. On V, define the addition as
a+a = a. ra Define the scalar multiplication as
=a
V scalar r .
Prove or disprove whether V becomes a vector space under these operations. (7 marks)
b) For each of the following cases, prove or or disprove whether S is a subspace of V. (7 marks)
i)
V = the vector space of all the n x n matrices. S= { A
IA
V.
E
V and At = A }
ii) V = any inner product space Let
Uo E
s = {u I U E V, < U , Uo > = I}
2. Let V be a vector space. a) Suppose
Prove that
{ Ul ,U2 ,
{ Uj
.u; } are linearly dependent if and only if at least one in
,U2 ,
.u.; } can be expressed as a linear combination of the others.
(5 marks)
b) Prove that any subset of V containing the zero vector must be linearly dependent. (2 marks)
c) Let u and v be non - zero vectors in V. Prove or disprove the following claim. u and v are linearly dependent => ( u + v ) and ( u - v ) are linearly dependent. (7 marks) If you are to prove this claim, you must give a rigorous proof. To disprove, you can give an example from any vector space of your choice.
589
UNIVERSITY OF MORA TUW A
Faculty of Engineering Department of Mathematics B. Sc. Engineering Level 2 - Semester 2 Examination: MA 2030 LINEAR ALGEBRA Time Allowed: 2 hours
2010 September 2010
ADDITIONAL
MATERIAL: None
INSTRUCTIONS
TO CANDIDATES:
This paper contains 6 questions and 5 pages.
Answer FIVE questions and NO MORE. This is a closed book examination.
Only the calculators approved and labeled by the Faculty of Engineering are permitted. This examination accounts for 70% of the module assessment.
Assume reasonable values for any data not given in or with the examination paper. Clearly state such assumptions made on the script.
If you have any doubt as to the interpretation of the wording of a question, make your own decision, but clearly state it on the script.
- 1-
MA2030
1. a) Let a be an object and define V = {a }. On V, define the addition as
a+a = a. ra Define the scalar multiplication as
=a
V scalar r .
Prove or disprove whether V becomes a vector space under these operations. (7 marks)
b) For each of the following cases, prove or or disprove whether S is a subspace of V. (7 marks)
i)
V = the vector space of all the n x n matrices. S= { A
IA
V.
E
V and At = A }
ii) V = any inner product space Let
Uo E
s = {u I U E V, < U , Uo > = I}
2. Let V be a vector space. a) Suppose
Prove that
{ Ul ,U2 ,
{ Uj
.u; } are linearly dependent if and only if at least one in
,U2 ,
.u.; } can be expressed as a linear combination of the others.
(5 marks)
b) Prove that any subset of V containing the zero vector must be linearly dependent. (2 marks)
c) Let u and v be non - zero vectors in V. Prove or disprove the following claim. u and v are linearly dependent => ( u + v ) and ( u - v ) are linearly dependent. (7 marks) If you are to prove this claim, you must give a rigorous proof. To disprove, you can give an example from any vector space of your choice.