Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 7
SC116 Algebraic Structures Autumn 2015
(Linear independence, basis, coordinates )
(1) Find three vectors in R3 that are linearly dependent and such that any two of them
are linearly independent.
Solution : Consider e1 , e2 and e1 + e2 . Clear
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 7
SC116 Algebraic Structures Autumn 2015
(Linear independence, basis, coordinates )
(1) Find three vectors in R3 that are linearly dependent and such that any two of them
are linearly independent.
(2) Are the vectors x1 = (1, 1, 2, 4), x2 = (2,
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 6
SC116 Algebraic Structures Autumn 2015
(Vector Spaces and subspaces)
(1) Let V be a set of real sequences (a1 , a2 , ., an , .) such that
V is a vector space over R.
P
i
a2i is finite. Prove that
(2) Let W1 and W2 be two subspaces of of a vect
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 10
SC116 Algebraic Structures Autumn 2015
(Inner Products)
(1) RLet V be the space of continuous functiontion from R to C. Show that hf  gi =
f (x)g(x)dx defines an inner product on V .
(2) Show that hA  Bi = Tr(A B) is an inner product on the
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 5
SC116 Algebraic Structures Autumn 2015
(Group Actions)
(1) Let G be he subgroup of S8 generated by (123)(45) and (78). This subgroup acts on
the set X = cfw_1, 2, ., 8. Compute the orbits and stabalizers of each of element of X.
(2) Consider a
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 6
SC116 Algebraic Structures Autumn 2015
(Vector Spaces and subspaces)
(1) Let V be a set of real sequences (a1 , a2 , ., an , .) such that
V is a vector space over R.
P
i
a2i is finite. Prove that
(2) Let W1 and W2 be two subspaces of of a vect
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 12
SC116 Algebraic Structures Autumn 2015
(Diagonalization,Eigenvalues, Eigenvectors)
(1) Find the characteristic polynomial, eigenvalues and bases for the eigenspaces of the
following matrices
1 2 2
3 1 1
1 2
2 1
a)
b)
c) 1 2 1 d)2 4 2
3 2
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 9
SC116 Algebraic Structures Autumn 2015
(Rank, Least squares )
(1) Show that if A and B are linear transformations on a finite dimentional vector space
V then
Rank(A + B) Rank(A) + Rank(B)
(2) Show that if A is m n and B is n p then
Rank(AB) =
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 4
SC116 Algebraic Structures Autumn 2015
(Groups and Subgroups)
1. .
i) S = cfw_(x, y) R R : x y is even.
reflexive : (x, x) S x R, since x x = 0 is even.
symmetric : (x, y) S x y is even y x is even (y, x) S.
transitive : (x, y), (y, z) S (x y)
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 1
SC116 Algebraic Structures Autumn 2015
(Groups and Subgroups)
(1) Find all the rotational symmetries of the cube.
b
a
A
D
C
B
c
C
D
B
A
Clearly, a = 3, b = 4 c = 2. In total there are 2 4 + 3 3 + 1 6 + 1 = 24 line
of symmetries. One can
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
lecture
symmetries of
Rotational
Symmetries

tetrahedron
regular
a
a
G
There
pg
and
3
these
(42)
are
of
symmetries
types of symmetries of type
M
total
of
and
identity
the
L
type
,
a
give
of
regular
a
symmetries
,
12
tetra
S2=e
tetrahedron
rotational
rota
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 4
SC116 Algebraic Structures Autumn 2015
(Quotient groups, Conjugacy classes, Lagranges theorem)
(1) Which are the subsets of R R are equivalence relations
i) cfw_(x, y)x y is and even integer
ii) cfw_(x, y)x y is rational
i) cfw_(x, y)x + y
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
lecture
symmetries of
Rotational
u
There
1
and
these
tetrahedron
regular
a
(42)
are
3
Symmetries

M
total
of
and
identity
the
a
give
of
4
L
type
types of symmetries of type
regular
a
symmetries
2
of
symmetries
.
,
12
tetra
.
3
rotation
by
about
S
L
anis
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 1
SC116 Algebraic Structures Autumn 2015
(Groups and Subgroups)
4. Show that if : G H is an isomorphism then x = (x), x G.
Solution : Let x G be of order n, then xn = 1.
Now, (x)n = (xn ) = 1. So, (x) divides n, (by Lagranges Theorem).
Now
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 2
SC116 Algebraic Structures Autumn 2015
(Cyclic groups, Permutation groups, Dihedreal group)
(1) Find all the generators of Z48 .
(2) Find all the subgroups of Z45 , giving a generator for each. Draw the lattice diagram.
n
(3) Ler p be prime an
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 3
SC116 Algebraic Structures Autumn 2015
(Group Isomorphism)
(1) Show that Z9 is isomorphic to Z6 .
(2) Explain why Z20 is not isomorphic to Z8 .
(3) Show that if : G H is an isomorphism then G is abelian iff H is abelian.
(4) Show that if : G H
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 1
SC116 Algebraic Structures Autumn 2015
(Groups and Subgroups)
(1) Find all the rotational symmetries of the cube.
(2) Find the order of
i) 2,6,10 in the additive group Z36 .
ii)5,7,7 in the multiplicative group Z12 .
iii)2 in the multiplicati