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

Winter 2016
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Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
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Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Balance analysis and light speed estimates
The Roofline Model
Set up a simple formalism to determine
upper performance limits on a given
architecture for a given numeric kernel
September 2013
Paralle
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
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Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
SC374 Set2
Harsh Vasoya (201501405)
Yash Kothari (201501413)
August 27, 2017
1
Taylor Polynomials
In this set, we are producing first, second and third order polynomials for the
given set of functions
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
SC374 Set2
Harsh Vasoya (201501405)
Yash Kothari (201501413)
August 27, 2017
1
Taylor Polynomials
In this set, we are producing first, second and third order polynomials for the
given set of functions
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Assignments
This document will be updated regularly. All assignments have to be submitted by a given
deadline.
Submission of assignments: Through Moodle. TAs will also explain in the Lab.
Make your re
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 8
SC116 Algebraic Structures Autumn 2015
(Linear transformations)
(1) Which of the following from R2 to R2 is a linear transformation
(a)T (x1 , x2 ) = (1 + x1 , x2 )
(b)T (x1 , x2 ) = (x2 ,
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 indep
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 indep
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. Pro
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)
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
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. Pro
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 matr
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)
Dhirubhai Ambani Institute of Information and Communication Technology
Algebraic Stuctures
MATH 116

Winter 2016
Tutorial 8
SC116 Algebraic Structures Autumn 2015
(Linear transformations)
(1) Which of the following from R2 to R2 is a linear transformation
(a)T (x1 , x2 ) = (1 + x1 , x2 )
(b)T (x1 , x2 ) = (x2 ,
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
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 ar
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
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 inte
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
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
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
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 the
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 t