Linear Algebra 2 (MATS3204)
BSc, Physics - PHYS
Semester: Second Semester
Level: 300
Year: 2017
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F ACULTE DE SCIENCE
MATS3204 EXAM 2017
DEPARTMENT OF MATHEMATICS AND
COMPUTER SCIENCE COURSE CODE: MATS3204 D r . D o r C .
Answer question 1, question 2 and question 3 I or II.
1. Let V be a vector space over a field and S = {v1, v
2
,…,v
n
} be a basis of V.
a) Define the corresponding dual basis {, …,
n
} of S. 2pts
b) Let be the dual space V and f ϵ , f ≠ 0. Show that for all , there exist v
such that f(v) =
c) If V =ℝ
3
and S = {v1 = (1, 0, 1), v2 = (0, 1, -1), v3 = (1, 1, 1) a basic of ℝ
3
, find
the corresponding dual basis. 6pts
d) If V = ℝ
3
and W
=<(1, 0, -1) > find the annihilator of W
2) Let V be a vector space over a field and W a subspace of V.
a) Define the relation R on V by xRy⟺ x− W. Show that R is an equivalence
on V. 6pts.
b) if V = ℝ
3
and W <(1, 2, 1), (1, -1, 1)>, describe the element (2, -1, 2) + W of
ℝ
3
/W. 5pts
c) Let W = < (-1, 1, 2) > and define T: ℝ3/W → ℝ
2
by T ([x, y, z + W]) = (x + y,
2y – z)
i) Show that T is well defined. 4pts
ii) Show that T is a linear transformation. 4 pts
iii) Find the kernel of T and hence determine whether T is a monomorphism. 5pts
3) Answer either question I or II
Let V be an inner product space over ℂ.
a) For x V, define ||x|| the norm of x. 1 pt
b) Prove that for all x, y V, ||x + y||≤||x||+||y||. 4pnts
c) A map d: V x V → ℂ is given by d(x, y) =||x – y||. Show that d(x,y) ≤ d(x,z) +
d(x,z) + d(z,y) for all x, y, z V. 3pnts
d) Let V = ℝ
2
and define the inner product < >: ℝ
2
x ℝ
2
→ ℝ by < (x
1
, y
1
), (x
2
, y2)
>:= 3x
1
x
2
+ 4
y
1y
2
.
Using the Gramm Schmidt Theorem, and thus inner product, obtain an
orthonormal basis for ℝ2 from the basis {v
1
= (2, 1), v
2
= (1, -1)}. 7pnts
II) Let V be an inner product space and T : V → V an endomorphism
a) Define T* the adjoint of T. 2pnts
b) Show that (T*)* = T. 5pnts
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1
c) Let {v
1
,…,v
n
] be an orthonormal basis of V and A the matrix of T with respect
to the basis. Show that A* =
is the matrix of T* with respect to the same basis.
4 pts
d) if V = ℂ3 and T : V → V is given by T(z
1
, z
2
, z
3
) := ((i – 1)z
1
+ z
3
, z
3
-iz
2
, z
1
+ (2
+ i)z
2
), find T* (z
1
, z
2
, z
3
). 4 pts.
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