Linear Algebra 2 (MATS3204)
BSc, Physics - PHYS
Semester: Resit
Level: 300
Year: 2017
1
UNIVERSITY OF BUEA
FACULTY OF SCIENCE
END-OF-SEMESTER EXAMINATION
DEPARTMENT: Mathematics INSTRUCTOR: Dr Dor Celestinc K
COURSE CODE: MAT 304 COURSE TITLE: Linear Algebra II
DATE: 12/09/2017 VENUE: Restau VI
INSTRUCTIONS: Answer all question, showing clearly all necessary steps in the answers.
1.
Let V be a vector space over a field
and V' the dual of V.
a)
If S = { v
1
, v
2
, . . . , v
n
} is a basis of V, define {
1
,
2
, . . . ,
n
} the corresponding dual basis of
S. (2 marks)
b)
Show that for all f ∈ V', f =
∑
(
)
(3 marks)
c)
Let U and W be subspaces of V and U° and W° their respective annihilators. Show that (U
+ W)° = U° ∩ W°. (5 marks)
2.
Let V be a vector space, W a subspace of V and T : V → V an endomorphism.
a)
Define a T-stable subspace of V. (1 mark)
b)
If V = ℝ
3
and T : ℝ
3
→ ℝ
3
is given by T(x, y. z) = (2x + y, x - y, 2x + y + 2z),
i)
Show that W =< (0,0,1) > is T-stable. (2 marks)
ii)
If W =< (0,0,1) >, show that {(1, 0,1) + W, (2,1, -1) + W} is a basis of the quotient space ℝ
3
/W
and hence find the matrix of the induced endomorphism
: ℝ
3
/W → ℝ
3
, where
[(x, y, z)
+ W] = T(x,y,z) + W, with respect to the basis {(l,0.1) + W, (2,1,-1) + W}. (5, 5 marks)
3.
Let V be an inner product space.
a)
Define an orthonormal set in V. (1 marks)
b)
Show that if S is an orthonormal subset of V, then S is linearly independent. (4 marks)
c)
Let V be the inner product space of all continuous functions on [0, 2] with inner product
defined by < f,g >=
∫
(
)
(
)
. Show that the subset S =
, where
(x) = sin kx,
is orthonormal. (5 marks)
d)
Prove that for all x,y ∈ V
, || x + y | | + | | x - y | | = 2 | | x | | + | | y | |. (4 marks)
4.
Let V be an inner product space and T : V →V an endormorphism.
a) Define T*, the adjoint of T . (2 marks)
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