Linear Algebra 2 (MATS3204)

BSc, Physics - PHYS

Semester: Second Semester

Level: 300

Year: 2017

Answers (200 )
THE UNIVERSITY OK BAMENDA
FACULTY OF SCIENCE
Department: Mathematics Course instructor: Dr Dor C . K.
Course Code: MATS 3204 Course Title: Linear Algebra II
Date: July 25, 2017 Time Allowed: 1 Hour 30 minutes
Instructions: Answer all questions, showing all necessary steps.
1. a) Let {v
1
= (1,-1), v
2
= (1,1)} be a basis of ℝ
2
and {w
1
= (1,0, -1), w
2
= (0,1,1), w
3
=(1,3,0)} be a basis of ℝ
3
. Find a basis for Hom
ℝ
(ℝ
2
, ℝ
3
).
(6 marks)
V
b) Let ∀ be a vector space over a field , W a subspace of V and {v
1
, v
2
, . . . , v
n
} a basis of V.
i) Define: the corresponding dual basis of {v
1
, v
2
, . . . , v
n
}, the annihilator of W. (2, 1 marks)
ii) Show that if dim V ≥ 2 and {u, v} linearly independent set in V then there exists a linear functional f ∈
, where
is the dual space of
V, such that f ( u ) = 1 and f ( v ) = 0. (3 marks)
iii) Show that the annihilator of W is a subspace of the dual space V'. (3 marks)
iv) If V = ℝ
3
and W = < (1, -1,3) >, find a basis for the annihilator of W . (5 marks)
2. a) Let V be a vector space over a field , W a subspace of V and V/W the quotient space.
i) Show that ∀ x + W, y + W e V/W, x + W = y + W ⇔ x - y ∈ W (4 marks)
ii) If V = ℝ
3
and W = < (2,1.3) > describe geometrically the element (3,1, -4) + W of ℝ
3
W. (3 marks)
b) Let W=< (3,1, -2) > and define a map T : ℝ
3
/W → ℝ
2
by T[(.
T
, y, z ) + W ] = ( x - 3y , 2 y + z ) .
i) Show that T is well-defined. (3 marks)
ii) Show that T is a linear transformation. (3 marks)
iii) Determine whether T is a monomorphism i.e. injective (3 marks)
GOOD LUCK
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