# Indian Statistical Institute, ISI MMATH 2020 PMA Solutions & Discussions

## Problem 1

Let be a finite dimensional vector space and let be a proper subspace of . Let be another subspace of such that , i.e. span and . Let be an invertible linear map such that . Which of the following statements is necessarily true

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

The map is injective, hence and maps onto : Let There exists such that But and there exists such that So,

is injective, hence So,

## Problem 2

Let be a finite dimensional real vector space, and let be a linear map such that Range. Which of the following statements in not necessarily true

(a) ;

(b) ;

(c) is an eigenvalue of ;

(d) All eigenvalues of are equal to .

## Solution:

Consider given by

Then but

## Problem 3

Consider the vector space equipped with the Euclidean metric define by

Let be a proper subspace of . Which of the following statements is necessarily true

(a) is closed.

(b) is open.

(c) is not closed.

(d) is neither closed nor open.

## Solution:

Let be a basis in , such that is a basis in Consider the linear map

given by is continuous. Then is closed.

## Problem 4

Let be a real matrix. If , let denote the cofactor of the entry , for . Let denote the matrix whose -th entry is . Suppose the rank of is . What is the rank of

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

Each element of is (up to sign) a minor of the matrix . Since the rank of is equal to 3, all minors of are equal to and has rank zero.

## Problem 5

For , the determinant of the permutation matrix

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

By induction we check that the determinant is equal to If then the determinant is equal to Assume the result is true for Expanding the determinant of the permutation matrix by the first row we get the answer

## Problem 6

Let denote the vector space of all matrices over the field of real number i.e.

Let be the subspace defined by

Then the dimension of is

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

The generic element of is of the form

## Problem 7

Let be a finite dimensional real vector space of dimension and let be a subspace of dimension . linear map from to is called a linear functional on . Which of the following statements is necessarily true

(a) There does not exist any linear functional on such that is the kernel of that linear functional;

(b) is the kernel of unique linear functional on ;

(c) is the kernel of a linear functional on ;

(d) There exist a non-zero linear functional on whose kernel strictly contains .

## Solution:

Let Then every element has a unique expansion

Define Then and

## Problem 8

Let denote the standard inner product on the vector space ,i.e.

for vectors . Let be a linear map such that

for all . which of the following statements is necessarily true
(a) .

(b) .

(c) .

(d) The map is onto

## Solution:

The map is injective, so it maps the basis of into a linearly independent subset of

## Problem 9

Let be a finite dimensional real inner product space and let be a linear map such that for all . Suppose is a proper subspace of such that Define a subspace of by

(a) is not contained in .

(b) is contained in .

(c) .

(d) is contained in

## Solution:

Let There exists such that For every we have and

So, and

## Problem 10

Let be a ring with unit such that for all . Which of the following statements is not neccessarily true
(a) for all ,

(b) for all ,

(c) is commutative,

(d) .

## Solution:

Let be the ring of all subsets of a set with operations

is an idempotent ring,

## Problem 11

Let be a nonempty set, and let be the power set of S, i.e. . Define a binary operation on by for . Which of the following statements is necessarily true

(a) is not a group as is not associative.

(b) is not a group as there is no identity.

(c) is an abelian group.

(d) is a non-abelian group.

## Solution:

With the operation , the power set is an abelian group:

## Problem 12

Let be ideals of a commutative ring . Define the set

Which of the following statements in not neccessarily true
(a) is an ideal of .

(b) .

(c) if is finite.

(d) if and is finite.

## Solution:

If with then and (c) fails.

## Problem 13

Let be an ideal of a commutative ring . Define the set

Which of the following statements is neccessarily true
(a) is an ideal.

(b) is not an ideal.

(c) .

(d) .

## Solution:

We check that is an ideal. Let Then for some We have

If then

If then In any case so and

If then for some and So,

## Problem 14

How many non-isomorphic group are there of order

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

There is an element of order 3 and an element of order 5. They generate cyclic Sylow subgroups and of . Let be the number of Sylow subgroups of Then

It follows that and is a normal subgroup of Similarly, is a normal subgroup of . Elements of commute with elements of and the order of is is cyclic.

## Problem 15

Suppose is a group and . which of the following statements is neccessarily true

(a) There exist a positive integer such that ;

(b) ;

(c) .

(d) None of the above statements is necessarily true.

## Solution:

If then

This proves By symmetry,

## Problem 16

Let be distinct subgroups of a finite abelian group . Define the subgroup by .
Which of the following statements is neccessarily true
(a) ;

(b) ,

(c) .

(d) .

## Solution:

Consider the mapping given by

is well-defined. Indeed, if

then and

is a homomorphism. Its kernel coincides with

So, the quotient of is an isomorphism.

## Problem 17

Let be a prime number and let denote the symmetric group on symbols. How many -Sylow subgroup are there in

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

Every Sylow subgroup of is a cyclic group. It contains elements of order Every element of order in is given by a permutation of There are such permutations, so the number of Sylow subgroups is

## Problem 18

Let be a commutative ring with unit and let

Which of the following statements is necessarily true
(a) Any prime of contains ,

(b) is not an ideal,

(c) is a prime ideal,

(d) .

## Solution:

Let be a prime ideal in . For any we have for some

The ideal is prime, hence one of the factors is in , i.e. and

## Problem 19

Let be twice continuously differentiable, and suppose Which of the following statements is necessarily true

(a),

(b) does not exist,

(c) ,

(d) .

## Solution:

Apply Taylor’s expansion:

applying L’Hopital’s rule twice

## Problem 20

Let be define by . Which of the following statements is true

(a) The sequence converges uniformly on ,

(b) The sequence converges pointwise on to a function such that has exactly one point of discontinuity,

(c) The sequence converges pointwise on to a function such that has exactly two points of discontinuity,

(d) The sequence does not converge pointwise on .

If then and

## Problem 21

Let be a continuous function such that

for all .Which of the following statements is necessarily false
(a) .

(b) .

(c) .

(d) .

## Solution:

Consider function is continuous on and for all

Since polynomials are dense in we deduce that In particular,

## Problem 22

Let be a continuous function such that for all . Which of the following statements is necessarily false

(a) for all ,

(b) does not exist,

(c) ,

(d) for all .

Let Let Then
and

## Problem 23

Let denote the set of all sequence such that for all . Define a map by

Which of the following statements is true
(a) The map is one-to-one and onto from to .

(b) The map is one-to-one and onto from to .

(c) The map is onto from and .

(d) The map is onto from to and .

## Solution:

Every real number has a binary expansion

The map is onto. We observe that

Equivalently,

## Problem 24

Let be a monotone increasing functions, and define by

for all . Suppose that for some , the limit , exists. Which of the following statements is necessarily false
(a) The sequence converges pointwise on .

(b) The sequence converges uniformly on .

(c) The limit exists.

(d) The function is unbounded on .

## Solution:

Let Let Find such that Then

is bounded.

## Problem 25

Let be set and let be a function. Let be a family of subsets of ,i.e. for all , Where is an index set.
Which of the following statements is not necessarily true

(a).

(b) .

(c) .

(d) .

## Solution:

Assume for all Let be non-empty sets such that Then

while

## Problem 26

Let be the set of all those nonnegative real numbers with the following property: if is a sequence of nonnegative real number such that , then we also have . Which of the following statements is true

( a) .

(b) .

(c) .

(d) .

.

## Solution:

By Cauchy-Schwarz inequality

The right-hand side is finite when i.e.

## Problem 27

Let be a finite set. Let be the power set of , i.e. the set whose elements are all subset of . Which of the following defines a metric on the power set

(a) .

(b) .

(c) .

(d) .

## Solution:

The metric is given by Indeed, from inclusion

it follows that

Further,

and

## Problem 28

The tangent line to the curve at the point has slope

(a) ,

(b) ,

(c) ,

(d) .

## Solution:

The slope of the tangent line is Differentiating the equation of the curve we get

At (1,2) we obtain

## Problem 29

Consider the following statements:

1. If and are convergent, then is convergent.
2. If is convergent and is absolutely convergent, then is absolutely convergent.
3. If for all , is convergent, and is bounded sequence, then is absolutely convergent, then is absolutely convergent.
Which of the following statements is true

(a) All of the statements are true,

(b) Statement is true but statement is false,

(c) Only statements and are true,

(d) Only statements and are true.

## Solution:

Statement (i) is false: consider the series The series is convergent by the alternating series test. Take Then both series and are convergent, but the series is divergent.

Statement (ii) is true. The sequence converges to zero, so it is bounded. Hence,

Statement (iii) is true by the same reasons: is absolutely convergent, sequence is bounded.

## Problem 30

Consider the metric space , Where the metric is define by for , and for . Let be a continuous function between metric space (Where is equipped with its usual metric). Which of the following statements is necessarily false
(a) The metric space is compact,

(b) The function is unbounded,

(c) The function is uniformly continuous,

(d) For any , the set is compact.

## Solution:

We prove that the metric space is compact. Indeed, let be an open cover of Let There exists such that If then and So, contains all points of except possibly a finite number of points. The remaining set of points can be covered by finite number of sets The space is compact. Continuous function defined on a compact space is bounded.
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