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PMS 2023 Mathematics Paper 2
(a) Show that if every element of a group G is its own inverse, then G is an Abelian
group.
(b) In a symmetric group of degree 3, give an example of two elements x, y such that
(x. y)² = x².y²
Show that if every element of a group G is its own inverse, then G is an Abelian
group.
In a symmetric group of degree 3, give an example of two elements x, y such that
(x. y)² = x².y²
(a) Show that normalizer of a subset H of a group G is a subgroup of G.
(b) Show that the centralizer of a subset H in a group G is a subgroup of G.
Show that normalizer of a subset H of a group G is a subgroup of G.
Show that the centralizer of a subset H in a group G is a subgroup of G.
(a) Define an integral domain. If p is a prime number, then show that ring of integers
mod p is an integral domain.
(b) Let F be a field of real numbers. Then set of all real valued functions whose nth
derivative exist for n = 1,2,..., is a subspace of all real valued continuous function on
[0,1].
Define an integral domain. If p is a prime number, then show that ring of integers
mod p is an integral domain.
Let F be a field of real numbers. Then set of all real valued functions whose nth
derivative exist for n = 1,2,..., is a subspace of all real valued continuous function on
[0,1].
(a) If G is a non-Abelian group of order 6, then show that G is isomorphic on to S₃.
(b) Let G₁ be the group of positive real numbers under multiplication and G₂ the group of
all real numbers under addition. Define : G₁ → G₂ by $$\phi(x) = \log_{10} x$$. Show that $$\phi$$ is a
bijective homomorphism.
If G is a non-Abelian group of order 6, then show that G is isomorphic on to S₃.
Let G₁ be the group of positive real numbers under multiplication and G₂ the group of
all real numbers under addition. Define : G₁ → G₂ by $$\phi(x) = \log_{10} x$$. Show that $$\phi$$ is a
bijective homomorphism.
(a) Let X be any infinite set, and let the set / consist of the empty set $$\phi$$ together with all
the subsets of X whose complements are finite. Show that / is a topology on X.
(b) Define a metrizable topological space. Give an example of a topological space which
is not metrizable.
Let X be any infinite set, and let the set / consist of the empty set $$\phi$$ together with all
the subsets of X whose complements are finite. Show that / is a topology on X.
Define a metrizable topological space. Give an example of a topological space which
is not metrizable.
(a) Let (X, <.,.>) be an inner product space. Show that
| < x,y > | < ||x||||y|| for all x, y ∈ X, where the equality sign holds if and only if {x,y} is
a linearly dependent set.
(b) Show that the space C[a, b] is not an inner product space
Let (X, <.,.>) be an inner product space. Show that
| < x,y > | < ||x||||y|| for all x, y ∈ X, where the equality sign holds if and only if {x,y} is
a linearly dependent set.
Show that the space C[a, b] is not an inner product space
(a) Show that the following can happen for 2x2 matrices A and B.
(1) A² = 0 even though A ≠ 0.
(2) AB + BA.
(b) Suppose that T: R⁴ → R³ is a linear transformation such that
T([1,0,-1,2]) = [2,1,0] and T([1,1,-2,0]) = [-1,2,4].
Compute T'([-2, -5,7,6]).
Show that the following can happen for 2x2 matrices A and B.
(1) A² = 0 even though A ≠ 0.
(2) AB + BA.
Suppose that T: R⁴ → R³ is a linear transformation such that
T([1,0,-1,2]) = [2,1,0] and T([1,1,-2,0]) = [-1,2,4].
Compute T'([-2, -5,7,6]).
Find the eigen values and the corresponding eigen vectors of
A =
[ 1 2 ]
[2 3 ]
Any two eigen vectors corresponding to two distinct eigen values of orthogonal
matrix are orthogonal.
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