Linear Algebra 2 (MATS3204)
BSc, Physics - PHYS
Semester: Second Semester
Level: 300
Year: 2018
1
REPUBLIC OF CAMEROON REPUBLIQUE DU CAMEROUN
Peace-Work-Fatherland Paix-Travaille-Patrie
MINISTRY OF HIGHER MINISTERE DE L’ENSEIGNEMENT
EDUCATION superieure
THE UNIVERSITY OF BAMENDA UNIVERSITE DE BAMENDA
FACULTY OF SCIENCE FACULTE DE SCIENCE
P.O. Box 39 BAMBILI
TEL: (+23)22 81 63 50
END OF SEMESTER EXAMINATION 2017/2018 SESSION
DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
COURSE CODE: MATS 3204 COURSE TITLE: LINEAR ALGEBRA II
DURATION: 3 HOURS
INSTRUCTIONS: ANSWER ALL QUESTIONS
1. Let V be a vector space over a field , V' the dual space and { v
1
, v
2
, . . . , v
n
} a basis of V.
a) Define { v'
1,
v'
2
, . . . , v'
n
}, the dual basis to the basis { v
1
, v
2
, . . . , v
n
}. (1 marks)
b) Show that if f ∈ V' and f ≠0 then ∀ ∈ , ∃ v ∈ V such that f ( v ) = . Hence deduce that every non-zero linear
functional is surjective. (4, 1 marks)
c) Find the dual basis corresponding to the basis
{v
1
= (1, 0, 1), v
2
= (0, -2, 1), v
3
= (1,1,0)} of ℝ
3
. (6 marks)
d) Find a basis of the annihilator of the subspace W =< (2, 1, 0), (4, 0, 3) > of ℝ
3
. (3 marks)
2. Let V be a vector space over a field and T : V → V an endomorphism.
a) Define a T-stable subspace W of V . (1 mark)
b) Let x be an eigenvector of T. Show that the subspace W = < x > is T-stable. (3 marks)
c) If V =ℝ
3
and T : ℝ
3
→ ℝ
3
is given by T(x, y, z) = (2x — y + 3z, y + 5z, 2z — 3y), find
i) a subspace W of ℝ
3
that is T-stable. (4 marks)
ii) Show that the set {(2,1,1) + TV, (0, —1,1) + TV}, where W is from (i), is a basis of the quotient space R
3
/TV.
(5 marks)
iii) Using the basis {(2,1,1) + W, (0, -1,1) + W}, Find the matrix of the induced map
: ℝ
3
/W → ℝ
3
/W, where
[(x,
y, z) + W] = T(x, y, z ) + W with respect to the basis {(2,1,1) + W, (0, -1,1) + W} of . ℝ
3
/W (5 marks)
3. Let V be an inner product space and W a subspace of V.
a) Define each of the following: an orthonormal set of vectors in V, the orthogonal complement of W. (2, 1 marks).
b) Show that an orthonormal set of vectors in V is linearly independent. (4 marks)
c) Show that for all x,y ∈ V
i) ||x + y|| ≤ ||x|| + ||y|| (4 marks)
ii) < x, 0 > = 0. (3 marks)
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