Algebra

Subject Code : 2K6M6:13

PAGE 1

ALGEBRA

SEMESTER-VISubject Code : 2K6M6:13

Level : K

Unit : 1

Type : MC

Sub Unit : 1.1

1.A non-empty subset H of a group G is a subgroup of G iff a, b H implies

(a) a b-1 H

(b) ab H

(c) a-1b H

(d) b-1a-1 H

2.Let H be a non-empty finite subset of G. IF H is closed under the operation in G then H is a

(a) finite group of G(b) subgroup of G(c) subset of G (d) not a subgroup of G

3.The union of two subgroups (H and K) of a group G is a subgroup iff

(a) KCH

(b) HCK

(c) K H

(d) HCK4.Let A and B be two subset of a group G then AB is equal to

(a){ab/aA, bB}(b){ab/aB,bA}(c){ab/a (A,b(((d) {ba/bA,aB}

5.Let A and B be two subgroups of a group G then AB is a subgroup of G if

(a) AB BA

(b) BA AB

(c) AB = BA

(d) AB BA

6.If A and B are subgroups of an abelian group G then AB is

(a) an abelian group of G (b) a subgroup of G (c) not a group (d) not a subgroup

7.Ha is called

(a) Centre (b) normaliser (c) subgroup (d) Improper subgroups, of a in G


Level : K

Unit : 1

Type : VSA

Sub Unit : 1.1

1.Define subgroup.

2.What is the subgroup of (R, +)?

3.What is the subgroup of (C, +)?

4.What is the subgroup of (R*, .)?

5.What is the subgroup of (C*, .)?

6.Define centre of a group G.

7.Define normaliser of a in G.


Level : K

Unit : 1

Type : P

Sub Unit : 1.1

1.Let H be a subgroup of G. Then prove that

(a) The identify element of H is the same as that of G.

(b) Prove that for each aH the inverse of a in H is the same as the inverse of a in G.

2.Prove that a non-empty subset H of a group G is a subgroup of G iff a,b H implies ab-1H.

3.Prove that the union of two subgroups of a group G is a subgroup iff one is contained in the other.

4.If H and K are subgroup of a group G then. Prove that H(k is also a subgroup of G.

5.Prove that a subgroup of a cyclic group is cyclic.

6.Let G be a group and AG then prove that order of a is the same as the order of the cyclic group generated by a.

7. (a) Prove that any cyclic group is abelian.

(3)

(b) Let G be a group and a be an element of order nin G then prove that am = e iff n divides m. (4)

8. (a) Let G be a group and let a be an elements of order n in G then prove that the order of as where 0 |G| and |K|>|G| then

prove that H(k ({e}.


Level : K

Unit : 1

Type : MC

Sub Unit : 1.4

1.For any group G the normal subgroups are

(a) {e} and {N}(b) {e} and G

(c) {e} and R

(d) {G} and G-12.Any subgroup of a cyclic group is

(a) normal

(b) not normal (c) centre

(d) normaliser

3.Let N be a normal subgroup of a group G. Then G/N is a group under the operation defined

by Na Nb is equal to

(a) Nba

(b) Nac

(c) Nab

(d) Ncd


Level : K

Unit : 1

Type : MC

Sub Unit : 1.4

1.The altering group An is a subgroup of index

(a) 3

(b) 5

(c) 2

(d) 6, in Sn

2.3z is a normal subgroup of (z, +). The quotient group Z/3z is a group of order

(a) 2

(b) 3

(c) 4

(d) 5


Level : K

Unit : 1

Type : VSA

Sub Unit : 1.4

1.Define normal subgroup.

2.What is the normal subgroup of (Z,+)?

3.Define quotient group of G modulo N.

4.What is the order of the quotient group Z6/?

5.What is the order of the quotient group Z60/

6.Define isomorphism.


Level : K

Unit : 1

Type : P

Sub Unit : 1.4

1.Let N be a subgroup of G. Then the following are equivalent.

(i) N is a normal subgroup of G.(ii) aNa-1 = N for all aG.

(iii) ana-1 c N for all aG.

(iv) ana-1N for all nN and aG.

2.Let N be a normal subgroup of a group G. Then prove that G/N is a group under the operation defined by Na Nb = Nab.

3.Prove that isomorphism is an equivalence relation among groups.

4.Show that if a group G has exactly one subgroup H of given order, then H is a normal

subgroup of G.

5.M and N are normal subgroup of a group G such that M(N = {e}. Show that every element

of M commutes with every element of N.

Subject Code: 2K6M6: 13

Level : K

Unit : 1

Type : E

Sub Unit: 1:4

1.(a)Let N be a subgroup of G. Then the following are equivalent.

(8)

(i) N is a normal subgroup of G

(ii) aNa-1= N for all aG

(iii) aNa-1C N for all aG

(iv) ana-1 EN for all nN and aG

(b)Let N be a normal subgroup of a group G. Then prove that G|N is a group under the

operation defined by NaNb=Nab.

(7)

2.(a)Prove that isomorphism is an equivalence relation among groups.

(8)

(b)Prove that a subgroup N of G is normal iff the product of two right cosets of N is again

a right coset of N.

(7)

3.(a)State and prove Cayley's theorem

(9)

(b)Prove that any finite cyclic group of order n is isomorphic to (Zn,().

(6)


Level : U

Unit : 1

Type : E

Sub Unit: 1:4

1.(a)Show that if a group G has exactly one subgroup H of given order, then H is a normal

subgroup of G.

(8)

(b)M and N are normal subgroups of a group G such that M(N=={e}. Show that every

element of M commutes with every element of N.

(7)


Level : K

Unit : 1

Type : MC

Sub Unit : 1.5

1.A bijetive homomorphism is called

(a) epimorphism(b) monomorphism(c) isomorphism (d) homomorphism

2.If f : G ( G1 is an epimorphism then G1 is called

(a) a homomorphic image of G

(b) isomorphism

(c) monomorphism

(d) ephimorphism

3.A homomorphism of a group to itself is called

(a) ephimorphism (b) monomorphism (c) endomorphism(d) isomorphism


Level : U

Unit : 1

Type : MC

Sub Unit : 1.5

1.Any homomorphism f is a monomirphism iff kerf is

(a) G

(b) G1

(c) {e}

(d) H

2.Any two cyclic groups with the same number of elements are

(a) isomorphic

(b) monomorphic(c) ephimorphic(d) endomophic


Level : K

Unit : 1

Type : VSA

Sub Unit : 1.5

1.Define automorphism of a group G.

2.Define inner automorphism of the group G.

3.Define homomorphism of the group G.

4.Define monomorphism.

5.Define kernel of f.

6.Define epimorphism.


Level : K

Unit : 1

Type : P

Sub Unit : 1.5

1.Prove that the number of automorphism of a cyclic of a cyclic group of order n is ((n).

2.Prove that for any group G,

(i) Aut G is a group under composition of functions.

(ii) I(G) is a normal subgroup of Aut G.

3.Let f:G( G1 be a homomorphism. Then prove that the kernal k of f is a normal subgroup of G.


Level : U

Unit : 1

Type : P

Sub Unit : 1.5

1.Let G be a finite abelian group of order n and let m be a positive integer relatively prime to n. Then prove that f:G(G defined by f(x)=xm is an automorphism of G.

2.Prove that any homomorphic image of a cyclic group is cyclic.


Level : K

Unit : 1

Type : E

Sub Unit: 1:5

1.(a)Prove that for any group G

(i) Aut G is a group under composition of functions.

(ii) I(G) is a normal subgroup of Aut G

(8)

(b)Let f: G ( G' be a homomorphism. Then prove that Kernal k of f us a normal subgroup of G

(7)

2.(a)State and prove fundamental theorem of homomorphism for groups.

(8)

(b)Prove that the number of automorphisms of a cyclic group of order n is ((n).

(7)Subject Code: 2K6M6: 13

Level : U

Unit : 1

Type : E

Sub Unit: 1:5

1.(a)Let G be a finite abelian group of order n and let m be a positive integer relatively prime to n, then prove that f:G(G defined by f(x)=xm is an automorphism of G.

(8)

(b)Prove that any homomorphic image of a cyclic group is cyclic.

(7)


Level : U

Unit : 2

Type : MCQ

Sub Unit : 2.1

1.(2z, + .) is a:

a) ring

b) Boolean ring c) null ring

d) Zero ring

2.Example of a Boolean ring is:

a) (P(s),(,()

b) (R + .)

c) (P(s),(,U)

d) (C,+ .)

3.In Ring R, (-a) (-b) =

a) (-a) b

b) a(-b)

c) (ab)

d) ab

4.In a Boolean ring v aR

a) a2 = a

b) a2 = a + a

c) a2 = 1

d) a2 = 0


Level : K

Unit : 2

Type : VSA

Sub Unit : 2.1

1.Define a ring.

2.Define ring of guassian integer.

3.Define ring of quaternions.

4.Define null ring.

5.Define boolean ring.

6.Write one of the properties of rings.


Level : U

Unit : 2

Type : VSA

Sub Unit : 2.1

1.Give an example for ring.

2.Give an example for Boolean ring.

3.Define a Commutative ring.

4.Define a ring with identity.


Level : U

Unit : 2

Type : PA

Sub Unit : 2.1

1.P.T. R={a+b (2 / a,bZ} is a ring with respect to usual addition and multiplication.

2.P.T. guassian integers is a ring under usual addition and multiplication.

3.S.T. (Z ( () is a ring where a (b = a+b-1 and a b = a + b - ab.

4.Show that (2z+*) is a ring where + is usual addition and * is given by a*b = ab.

5.If R is a ring such that a2 = a VaR. Prove that

(i) a + a = 0

(ii) a + b = 0 ( a = b

(iii) a + a = 0.


Level : U

Unit : 2

Type : E

Sub Unit : 2.1

1. a)Prove that R={a+b 2 / a,b Q} is a ring w.r.t. usual addition and $ multiplication.

b)If R is a ring such that( a2=a Va R prove that

(i) a + a = 0

(ii) a + b = 0 ( a = b

(iii) ab = ba.

(9+6)

2. a)Show that (2z,+,*) is a ring where + is usual addition and * is given by a * b = 1/2 ab.

b)State and prove the properties of rings.

(9+6)


Level : U

Unit : 2

Type : MCQ

Sub Unit : 2.2

1.A non commutative ring is

a) (z + .)

b) (Q + .)

c) (R + .)

d) (M2(R), +,.)

2.(2Z, + .) is a ring

a) with identity b) without identity c) non-commutative ringd) I.D

3.An infinite non commutative ring with identity is

a) M2(R)

b) Z4

c) 2Z

d) R


Level : K

Unit : 2

Type : VSA

Sub Unit : 2.2

1.Give an example for commutative ring.

2.Give an example for non-commutative ring.

3.Give an example for finite commutative ring.

4.Give an example for a ring with identity.

5.Give an example for a ring without identity.

6.Define a unit element in R.

7.Define skewfield

8.Define field

9.Define zero divisors of a ring.


Level : K

Unit : 2

Type : PA

Sub Unit : 2.2

1.Let R be a ring with identify then prove that set of all units in R is a group under multiplication.

2.In a skewfield R, Prove that

(i) ax = ay, a 0 ( x = y

(ii) xa = ya a 0 ( x = y

(iii) ax = 0 (> a = 0 or x = 0.

3.Prove that a finite commutative ring R without zero divisors is a field.

4.Prove that Zn is a field iff n is prime.

5.Prove that a ring R has no zero divisors iff consultation law is valid.


Level : U

Unit : 2

Type : PA

Sub Unit : 2.2

1.State and prove properties of rings.

2.Prove that M2 (R) is a non commutative ring.

3.Let F denote the set of functions f:R(R, defined by (f+g) x=f(x) + g(x) and f.g. = fog then

prove that (F,+ .) is non commutative ring.


Level : U

Unit : 2

Type : E

Sub Unit : 2.2

1.a)Prove that M2 (R)is a non commutative ring

(9+3)

b)Prove that in a ring with identity, the identity is unique

c)Write down the identity element in Z, M2(R) & C.

(3)

2.Show that (RxR, + .) is a

a)Commutative ring with identity where the + and . defined as

(a,b) + (c,d) = (a+c, b+d) &

(a,b) + (c,d) = (ac, bd)

b)Give an example for a ring which has no identity and prove it.

(10+5)


Level : K

Unit : 2

Type : MCQ

Sub Unit : 2.3

1)Let R be a ring with identity, then the element UR is a unit ifa) U has multiplicative inverse b) U + 0 = 0

c) U2 = U d) U2 = 1

2.A ring R is a division ring if

a) It forms a group under multiplication

b) It has the unit element

c) It has the unit element and cancellation laws hold

d) Its all non zero element form a group under multiplication.

3.In a skewfield R, ax=0 if

a) a 0 & x 0b) a = 0 & x 0 c) a 0 & x = 0 d) a = 0 or x = 0Subject Code : 2K6M6:13

Level : U

Unit : 2

Type : MCQ

Sub Unit : 2.3

1.The units in (Z + .) are

a) 1 only

b) 1 only

c) 1 & -1

d) all integers

2.(M2(c) + .) is

a) neither commutative nor a skewfiledb) Commutative but not a skewfieldc) Not commutative but a skewfield

d) A field

3.In the ring Z6, the one of the divisors is:

a) 2

b) 9

c) 8

d) 11

4.If a ring R has no zero divisors then:a) R is a skewfield =

b) R need not be a skewfield

c) cancellation law is valid in R

d) all the non zero elements of are unity


Level : U

Unit : 2

Type : VSA

Sub Unit : 2.3

1.Give an example of a field.

2.Give an example of a skewfield.

3.Give an example of skewfield but not a field.

4.What are the zero divisors in (RxR,+.)?5.What are the zero divisors in Z12?

6.What are the zero divisors in M2(R)?


Level : U

Unit : 2

Type : PA

Sub Unit : 2.3

1.Prove that (Z,+ .) is not a field.

2.Prove that F={a+b 2/ a,bEQ) is a field.

3.Let R be a ring and let a be fixed element in R. Let Ia = {xR/ax=0} Show that Ia is a

subring of R.

4.Check whether the map f:c(C defined by f(z)= ( is a ring isomorphism or not.


Level : K

Unit : 2

Type : E

Sub Unit : 2.3

1.a)Let R be a ring with identity. Then prove that set of all units in R is a group under multiplication.

b) In a skewfield prove that ax = ay, a 0 ( x = y and ax = 0 a = 0 or x = 0.

(8+7)

2.a)Prove that R has no dovisors iff cancellation law is valid in R.

(7+8)

b)Prove that a finite commutative ring R without zero divisors is a field.


Level : U

Unit : 2

Type : MCQ

Sub Unit : 2.4

1.(Z + .) is:

a) a skewfield b) a field

c) an I.D d) a non commutative ring

2.The only Idempotent elements of an ID are

a) 1 only

b) 0 only

c) 0+1

d) all non zero elements

3.Z9 is a ring of characteristic

a) 6

b) 3

c) 2

d) 9

4.(P(s),(,() is a ring of characteristic a) 2

b) 3

c) 4

d) 0

5.Z is a ring of characteristic

a) 2

b) 0

c) 4

d) n

6.The characteristic of a boolean ring is:a) 2

b) 0

c) 3

d) even number

7.{0,2} is a subring of:

a) Q

b) Q

c) R

d) Z48.A subring of field which is not a field is:a) Q

b) R

c) Z

d) M2(R)

9.A subring without identity of a ring with identity is:a) Q

b) 2Z

c) Z

d) M2(R)


Level : K

Unit : 2

Type : VSA

Sub Unit : 2.41.Define an Integral Domain.

2.Define the characteristic of a ring.

3.Define subring of a ring R.

4.Define subfield of a field F.

5.What is the characteristic of an I.D.?

6.What is the characteristic of any field?


Level : U

Unit : 2

Type : VSA

Sub Unit : 2.4

1.Give an example for an I.D.

2.Give an example for a finite commutative ring with identity which is not an i.D.

3.What is the characteristics of the ring Zn?

4.What is the characteristics of the Boolean ring?5.Give an example of ring of characteristic zero.

6.Give an example of subring of a ring M2(R).

7.Give an example of subfield of a field R.

8.Give an example for subring but not an ideal.


Level : K

Unit : 2

Type : PA

Sub Unit : 2.4

1.Prove that Zn is an I.D iff n is prime.

2.Prove that any filed is an I.D. but not conversely.

3.Prove that any unit in R cannot be a zero divisor but not conversely.

4.Prove that if R is an I.D iff the set of non zero elements in R is closed under multiplication.

5.Prove that R is an I.D. iff cancellation law is valid in R. Prove that any finite I.D is a field.

6.Prove that the characteristic of an I.D. either 0 or prime.

7.Prove that an I.D. of characteristic P, the order of every element in the additive group is P.

8.Prove that the necessary and sufficient condition for subring of R.

9.Prove that inter section of two subrings of ring is a subring but need not be true for union.

10.Prove that union of two subrings of a ring is again a subring iff one is contained in the other.

11.Prove that necessary and sufficient condition for subfield of a field R.


Level : K

Unit : 2

Type : E

Sub Unit : 2.4

1. a)Prove that Zn is an I.D iff n is prime.

(7+8)

b)Prove that any field is an I.D. but not conversely.

2. a)Prove that any finite I.D. is a field

(7+8)

b)Prove that the characteristic of an I.D. is either O or prime.

3. a)Prove that union of two subring of a ring is again a subring iff one is contained in the other.

b)Let R be a ring and let a be fixed element of R.

(9+6)

Let Ia = {xR / ax=0}. Show that Ia is a subring of R.


Level : K

Unit : 2

Type : MCQ

Sub Unit : 2.5

1.An onto homomorphism is called

a) isomorphism b) epimorphism c) endomorphism d) monomorphism

2.A 1-1 homomorphism is calleda) monomorphism b) endomorphism c) epimorphism d) isomorphism

3.A homomorphism of a ring to itself is called

a) monomorphism b) endomorphism c) epimorphism d) Isomorphism

4.An isomorphism of a ring is

a) an ideal

b) homomorphism c) subring

d) PID

5.Every ideal of a ring R is a

a) ring of R

b) Quotient ring c) subring of R d) commutative ring

6.Let R be a ring and I be an ideal of R then R is ring with identity implies R/I is

a) Ring with identity b) an I.D

c) field

d) non commutative ring

7.If f is an isomorphism then f(ab)

a) f(a) (f(b)

b) f(a) + f(b)

c) f(a)f(b)

d) af(b)

8.The mapping f:c(C defined by f(z) = ( is

a) only a 1-1 mapping b) only an onto c) only a homomorphism d) an isomorphism

9.A homomorphism f from a ring R+OR1 is an isomorphism if

a) f is bijection b) f is 1-1

c) f is onto d) f(ab) = f(a)f(b V a,bER


Level : U

Unit : 2

Type : MCQ

Sub Unit : 2.5

1.Q is a

a) PID

b) not an ideal

c) not a principal ideal d) not a PID

2.An ideal of Z is

a) 2Z

b) Q

c) M2 (R)

d) {0,1}

3.A subring but not an ideal is

a) 2Z

b) Z

c) Q

d) M2(R)

4.If f: R( R defined by f(a)=2a then

a) homomorphism b) an isomorphism c) 1-1 only

d) onto only


Level : U

Unit : 2

Type : VSA

Sub Unit : 2.5

1.Give an example of an ideal of a ring.

2.Give an example of Principal ideal.

3.Give an example of P.I.D.


Level : K

Unit : 2

Type : VSA

Sub Unit : 2.5

1.Define Quotient ring.

2.Define homomorphism of rings.

3.Define kernal of homomorphism of rings.

4.Define monomorphism of ring.

5.Define epimorphism of ring.

6.Define endomorphism of ring.

7.State fundamental theorem of homomorphism of rings.

8.Define an ideal of a rig R.

9.Define principal ideal.

10.Define principal ideal Domain.


Level : K

Unit : 2

Type : PA

Sub Unit : 2.5

1.Prove that a field has no proper Ideals.

2.Let R be commutative ring with identity. Prove that R is a field iff R has no proper idelas.

3.State and prove fundamental theorem of homomorphism of rings.

4.Prove that sum of two ideals is an ideal.

5.Prove that undersection of two ideals is an ideal.

6.Prove that every idela is a subring but not conversely.

7.Let f:R( R1 be a homomorphism. Let K be a kernal of f. then prove that K is an idela

8.Let f:R(R1 be an isomorphism then prove that (1) R is a ring with identity implies R is a

ring with identity. (2) R is a field ( R1 is a field

9.Let f:R(R1 be an isomorphism then prove that R is a commutative ring with identity implies

R1 is a commutative ring with identity.


Level : K

Unit : 2

Type : E

Sub Unit : 2.5

1. a)State and prove fundamental theorem of rings.

(10+5)

b)Let f:R(R1 be homomorphism. Then prove that f is 1-1 iff kerf = {0}

2. a)Prove the intersection of two ideal is an ideal.

(9+6)

b)Prove that every ideal is a subring but not conversely.

3. a)Define Quotient ring and prove that it is a ring

(10+5)

b)Prove that a field has no proper ideals.


Level : U

Unit : 3

Type : MC

Sub Unit : 3.1

1.With usual notations, C(R) is:

a) A group onlyb) a field only c) a vector spaced) Not a vector space

2.R ( R with usual addition of ordered pairs and scalar multiplication defined by

((x,y) = ((x, (2y), (ER is:

a) A field only

b) a vector space over R

c) not a vector space

d) an inner product space

3.R ( R with usual addition of ordered pairs and the scalar multiplication defined by

( (x,y) = (( x,y), ( ER is:

a) A vector space over R

b) not a vector space

c) a field only

d) a skew field only

4.R ( R with usual addition of ordered pairs and the scalar multiplication defined by

((x,y)=(x,( y)is:a) Not a vector spaceb) a vector space over R c) a group onlyd) a field only

5.R ( R with usual addition of ordered pairs and the scalar multiplication defined by ((x,y) = 0 is:a) a vector space over R b) not a vector space c) a field only d) a skew field only

6.R (C) is:a) a ring only b) a field only c) also a vector space d) is not a vector space

7.R ( R (C) is:a) A field only b) a ring only c) not a vector space

d) a vector space


Level : K

Unit : 3

Type : VSA

Sub Unit : 3.1

1.If V is a vector space over F, how do you call the elements of V and F?

2.If V is a vector space over F, prove that ( = for all ( EF

3.If V is a vector space over F, Prove that 0v = for all v V

4.If V is a vector space over F, Prove that (-() v = - ((v) for all ( F & v V

5.If V is a vector space over F, Prove that ((-v)= - ((v) for all (F, vV.


Level : U

Unit : 3

Type : VSA

Sub Unit : 3.1

1.Consider RxR with usual addition. Define the scalar multiplication as ( (a,b) = (0,0).

Which axiom fails to satisfy w.r.to scalar multiplication for RxR to be a vector space over R?

2.Let RxR = {(a,b) / a, bR}. Addition and scalar multiplication are defined by (a1, b1) + (a2, b2) = (a1+a2) + (b1+b2) = and ( (a,b) =(a,(b) respectively. Is the axiom ((+) (a,b) = ( (a,b) + (a,b) satified.

3.Consider RxR with usual addition and multiplication defined by ((a,b) = Prove that ((+) (a,b) ((a,b)+ (a,b)=(0,(b).

4.Let V be any vector space over a field F. Prove that ((u-v) = (u - (v5.Let V be any vector space over a field F. Prove that (u=(v and ( 0 ( u = v.

6.Let V be any vector space over a field F. Prove that (u=u and ( 0 ( ( = .

7.Is R closed w-r-to scalar multiplication where the scalars are taken from C? Justify.

8.Consider RxR with usual addition and the scalar multiplication defined by

((x,y) = ((x,y). Check whether the axiom ((+) (x,y) = ((x,y) + (x,y) satisfies or not.

9.Consider R ( R with usual addition and the scalar multiplication defined by

( (x,y) = ((x, (2y). Is(((+) (x,y) = ((x,y) + (x,y)?


Level : K

Unit : 3

Type : PA

Sub Unit : 3.1

1. a)Define a vector space

b)If V be a vector space over a field F, Prove that ( = V(F

2.Let V be a vector space over a field F, Prove that

(i) (-() v = ((-v) = - ((v) for all (F & v V

(ii) (v=0 ==> (=0 or v=

(iii) ((u-v) = (u - (v


Level : U

Unit : 3

Type : PA

Sub Unit : 3.1

1.Prove that M2 (R) of all 2x2 matrices is a vector space over R under matrix addition and

scalar multiplication defined by

2.Let V = . Prove that V is a vector space with respect to usual addition

of differential equations and the scalar multiplication defined

by .

3.Let R+ be the set of all positive real numbers. Define addition and scalar multiplication as follows u + v = uv Vu, vER+ and (u = U(, V UER+ & (ER. Prove that R+ (R) is a vector space.

4.Let V= {a + b(2 + C(3 / a,b,c,,Q}. Define addition and multiplication as follows (a1,b1,(2 + C1(3) + (Q2 + b2,(2 + C2(3) = (a1 + a2) + (b1 + b2)(2 (C1 + C2)(3 and

( (a+b(2+C(3) = (a+ (b(2+(C(3 Prove that V is a vector space over Q.

5.Let V = {0,1,2,x+1,x+2,2x+1,2x+2,x,2x} CZ3 [x] Prove that V is a vector space over Z3.

6.Let U and W be vector spaces over the same field F. Let V = {(u,w) / u U, w W}.

Show that V is a vector space over F with addition and scalar multiplication defined by

(u,w) + u1,w1) = (u+u1, w+w1) and k (u,w) = (ku,kw).


Level : K

Unit : 3

Type : E

Sub Unit : 3.1

1.a)Define a vector space over a field F.

b)Consider RxR with usual addition. Define the scalar multiplication as ((a,b) = (0,0).

Check whether all the axioms w-r-to scalar multiplication satisfy or not.

2.Let V be a vector space over a field F. Then prove the following

(i) (( = ( for all ( E F

(ii) Ov = ( for all vEV

(iii) (-() v = ((-v) = - ((v) for all (F and vV(iv) (v=(( (=0 or v=((v) ((u-v) = (u-(v for all (EF & u,vEV.


Level : U

Unit : 3

Type : E

Sub Unit : 3.1

1.a)Let R+ be the set of all positive real numbers. Define addition and scalar multiplication

as follows u + v = uv, V u,vR+ and (u = u( for all uER+ & (ER. Prove that R+ is a vector

space over R.

b)Prove that R ( R with addition defined by (a,b) + (c,d) = (a-c, b-d) and usual

scalar multiplication ( (a,b) = ((a, (b) is not a vector space over R.

2.Prove that

(i) V = {a + b(2 + C(3 / a,b,c,,Q} is a vector space over Q with respect to usual addition & scalar multiplication.

(ii) Z is not a vector space over Q.


Level : U

Unit : 3

Type : MC

Sub Unit : 3.2

1.{0} and V(R) are

a) not the subspaces of V

b) proper subspaces of V

c) trivial subspaces of V

d) subfield of V

2.W = {(a,0,0) / aR} is a subspace of

a) R4 (R)

b) R3(R)

c) R2 (R)

d) R(R)

3.W = {(x,y,z) R3 / lx + my + nz = 0} is a subspace of

a) R(R)

b) R2(R)

c) R3(R)

d) R4(R)

4.If A and B are subspaces of V, then A(B is: a) A subspace of Vb) not a subspace of Vc) contained in A-B d) contained in B-A

5.If A and B are subspaces of V such that V = A ( B, then

a) A(B = {1}

b) A(B = {0} c) A(B CA-B d) A(BC B-A

6.If A and B are subspaces such that V = A ( B, then every element of V can be

uniquely expressed in the form

a) a-b

b) ab

c) , b(0 d) a+b (where aA, bB)

7.If A={(a,b,0) / a,bER} and B = {(0,0,C) / CER}, then V3 (R) equals.a) A ( B

b) A + B

c) A - B d) (AUB) (A(B)


Level : K

Unit : 3

Type : VSA

Sub Unit : 3.2

1.Define a subspace of a vector space V(F)

2.Write down trivial subspaces of V(F)

3.State the necessary and sufficient condition for a non-empty subset W of V to be a subspace

of V.

4.If W is a subspace of V, Prove that (u+ v W whenever u, vW & (, f.

5.When do you say that V is the direct sum of two subspaces of V?

6.Define quotient space.


Level : U

Unit : 3

Type : VSA

Sub Unit : 3.2

1.Prove that W = {(a,0,0)/a R} is a subspace of R3(R).

2.Prove that W = a 0 {a,b (R) is a subspace of M2(R)0 b}

3.Prove that W = {(ka,kb,kc)/k R} is a subspace of R3(R)

4.If A = {(a,0,0)/a R},b{(0,b,o)/b R}, is AUB a subspace of R3? Justify.

5.Show that {(a,0,c) / a,CR} is a subspace of R3(R)

6.Let A = {(a,b,0)}and B = {O,O,C)} where a,b,C,,R Prove that R3(R) = A ( B.


Level : K

Unit : 3

Type : PA

Sub Unit : 3.2

1.Let V be a vector space over F. A non-empty subset W of V is a subspace of V iff W is

closed w.r.to vector addition and scalar multiplication in V Prove.

2.Let V be a vector space over a field F & W be a non-empty subset of V. Prove that W

is a subspace of V iff u,vEW and (,(EF ==> (u+(v W

3.Let V be a vector space over F and W a subspace of V. Let V = {W + v/vV}. Prove that V

is a vector space F under W the following operations.

(i) (W+v1) + (W+v2) = W+v1+v2

(ii) ((W+v1) = W + (v1Subject Code : 2K6M6:13

Level : U

Unit : 3

Type : PA

Sub Unit : 3.2

1. (a)Let W = {(a,b,c) R3 / la + mb+nc=0} Prove that W is a subspace of R3.

(b)Let W={(a,0,0) R3/aR}. Prove that W is a subspace of R3.

2. (a)Prove that the intersection of two subspaces of a vector space is a subspace.

(b)Prove that the union of two subspaces of a vector space need not be a subspace.

3.Let A and B be subspaces of a vector space V. Prove that A(B = {0} if every vector vA+B

can be uniquely expressed in the form v=a+b where aA, bB.

4.If A and B are subspaces of V, Prove that A+B = {vV / v=a+b, aA, bB} is a subspace of V. Also prove that A+B is the smallest subspace containing A and B.

5.Prove that

a) {(a,0,C) / a,CR} is a subspace of R3.

b) {(x,y,z) / x2+y2+z2y>+z} is not a subspace of R3(R).

8.Let A = {(a,b,0) and B = {(0,0,C)} be the subspaces of R3(R).

a) Prove that R3 (R) = A(B

b) Prove that {(a,b,a+b)} is a subspace of R3(R).


Level : K

Unit : 3

Type : E

Sub Unit : 3.2

1.a)Let V be a vector space over F. A non-empty subset W of V is a subspace of V if W is

closed w.r.to vector addition and scalar multiplication in V Prove.

b)Prove that the intersection of two subspaces of a vector space is a subspace.

2.a)Let V be a vector space over F and W a subspace of V. Let V = {W+v / vV}. Prove that V

is a vector W W. Space over F under the following operations.

(i) (W + v1) + (W + v2) = W + v1 + v2

(ii) ((W + v1) = W + (v1

b)Prove that W={ka,kb,kc) / kER} is a subspace of R3.

3.a)Let V be a vector space over a field F. Prove that a non-empty subset W of V is a subspace

of V if u,V,,W and (,(EF( (u+(v w

b)Prove that W = / a,bER}is a subspace of M2(R)


Level : K

Unit : 3

Type : MC

Sub Unit : 3.3

1.Let W be vector spaces over the same field F. Then the mapping T : V( W is called a

linear transformation if for u,vEV and (,(EF,

a) T ((u + (v) = (T(u + (v)

b) T((u + (v) = (T((u + v)

c) T ((u + (v) = ((T(u + v)

d) T((u + (v) = (T(u) + (T(v)

2.Let V and W be vector spaces over the same field F. Then the linear transformation T:V(W

is a monomorphism if

a) T is 1-1 and intob) T is ontoc) T is 1-1d) T is a mapping from V to F

3.Then the linear transformation T: V(W is an epimorphism if T is

a) 1-1

b) onto

c) 1-1 and ontod) a mapping from V to itself

4.Let V and W be vector spaces over a field F. Then the linear transformation T:V( W

is an isomorphism of T is:a) 1-1

b) onto

c) 1-1 and ontod) a mapping from V to F5.A linear transformation T:V( F is called

a) A monomorphism b) an epimorphismc) an isomorphism d) a linear functional

6.Let V and W be vector spaces over a field F and T:V(W be a linear transformation.

Then the kernel of T is defined asa) {v/vEV and T(v)=0}

b) {v/vEV and T(v) =1}

c) {v/vEV and T(v)=v}

d) {v/vEV and T(v) (0}

7.Let V and W be vector spaces over a field F and T:V( be a linear transformation.

Then T is a monomorphism if:a) KerT=V

b) KerT={1}

c) KerT={0}

d) KerT={0,1}

8.Let V and W be vector spaces and T:V( W be an epimorphism. Then

a) V----- ( W b) V- ker T (W c) V = W + img(ker T) KerT d) V(W-Img(Ker T)

9.Let A and B be subspaces of V. Then

a)

b)

c) d)

10.The kernel of the linear transformation T:V4(R)( V4(R) (R )(V4(R) defined by

T{(X1, X2, X3, 0)} = (X1, 0, X3, 0) is

a) {(x1,x2,x3,0)} b) {(x1,x2,0,0)} c) {(0,x2,0,x4,)} d) {(x1,x2,x3,x4)}

11.If S={(1,0), (0,1)}, then

a) L(S) = R(R) b) L(S)=C(R)

c) L(S) = V2(R)d) L(S)=V3(R)

12.If S = , then

a) L(S) CM2 (R) b) L(S) M2 (R) c) L(S)C CM2 (R) d) L(S) V2(R)

13.S = {(2,0)} is:a) Linearly independent

b) linearly dependent

c) such that L(S)=V2 (R)

d) such that L(S) CV2 (R)

14.If S = , then

a) S is linearly independent b) S is linearly dependent c) L(S)(M2(R) d) L(S)CM2(R)


Level : K

Unit : 3

Type : VSA

Sub Unit : 3.3

1.Define linear transformation

2.Define monomorphism of vector spaces.

3.Define an epimorphism of vector spaces.

4.Define an isomorphism of vector spaces.

5.When do you say that two vector spaces are isomorphic to each other?

6.Define : linear functional

7.Define : trivial linear transformation

8.Define : identity linear transformation

9.State the necessary and sufficient condition for T:V(W to be a monomorphism

10.Define the kernal of the linear transformation T:V( W.

11.State fundamental theorem of homomorphism vector spaces.


Level : U

Unit : 3

Type : VSA

Sub Unit : 3.3

1.Prove that V3(R)( V3(R) defined by T(a,b,c) = (a,o,o) is a linear transformation.

2.Find the kernal of T : V4(R)( V4 (R)defined by T(X1,0,X3,0)= (x1,0,x3,0)

3.Find the kernel of T:V3(R)(V3(R) defined by T(a,b,c) = (a,b,0) Is T:V3(R)( R defined by

T (x,y,z) = x2+,y2+,z2, a linear transformation? How?


Level : K

Unit : 3

Type : PA

Sub Unit : 3.3

1.a) Let T:V ( W be a linear transformation. Prove that T(V) = {T(v) / vEV} is a subspace of W.

b) Let T:V ( W be a linear transformation. Prove that T is a monomorpshism iff kerT{0}

2.State and prove fundamental theorem of homomorphism of vector spaces.

3.Let V be a vector space over a field F and A &B be subspaces of V. Prove that

4.Let V and W be vector spaces over a field d F. Let L(V,W) = {f/f is a linear transformation from V to W}. Prove that L(V,W) is a vector space over F under addition and scalar multiplication defined by (f+g) (v) = f(v)+g(v) and ((f)(v)= ((f(v).


Level : U

Unit : 3

Type : PA

Sub Unit : 3.3

1.a) Let V be a vector space over a field F and W a subspace of V. Prove that T:V( V defined by T(v) = W+v is a linear transformation W.

b) Prove that T:V3(R)(V3(R) defined by T(a,b,c) = (a,0,0) is a linear transformation.

2.Prove that T:R2( R2 defined by T(a,b) = (2a-3b,a+4b) is a linear transformation.

3.Prove that T:V( Vn+1 (R) defined by T(a0,a1x+a2x2+... + anxn) = (a0,a1,... an) is a linear transformation where V is the set of all polynomials of degree n is:a) linearly dependent

b) linearly independent

c) both linearly dependent and independent

d) neither linearly dependent nor linearly independent

10.Let dim(V)=n then any set of vectors with mx-y

2.If equality is valid in schwartzs inequality then x+y are

a) L.I

b) L.D.

c) Zero d) opposite in sign

3.x+y2 = x2+y2 if =

a) x+y

b) x+y

c) 0

d) 1

4.X + y are orthogonal if =

a) o

b) x2+y2

c) 1

d)

5. ( x is:a) x

b) ( y c) ( x

d) ( x

6.The Schwartzs inequality is:a) x+y < x + y

b) || > x y

c) || < x y

d) x-y > x y


Level : K

Unit : 5

Type : VSA

Sub Unit : 5.2

1.Write any one of the properties of norm.

2.State schwartzs inequality.

3.State Triangles inequality.

4.Define distance between any two vectors.

5.Write any one of the properties of d(x,y)

6.Define orthogonal vectors.

7.Define an orthogonal set.

8.Define an orthonormal set.


Level : U

Unit : 5

Type : VSA

Sub Unit : 5.2

1.Find the norm of (1,1,1) in the inner product space V3(R).

2.Give an example for orthogonal set.


Level : K

Unit : 5

Type : PA

Sub Unit : 5.2

1.State and prove schwartzs inequality

2.State and prove Triangle inequality

3.Prove that any orthogonal set of non zero vectors in an inner product space V is Linearly independent.

4.Show that in any inner product space V ||x + y||2 x - y||2 = 2 (||x||2 + ||y||2).

5.State and prove the properties of the norm defined in an inner product space.

6.Show that if equality is valid in Schwartzs inequality then x & y are LD.


Level : U

Unit : 5

Type : PA

Sub Unit : 5.2

1.In an inner product space defined the distance between any two vectors x & y by d(x,y) = ||x-y|| show that

a) d(x,y)( 0

b) d(x,y) = 0 iffc) d(x,y) = d(y,x)d) d(n,z) + d(z,y)

2.Find an orthogonal basis containing the vectors (1,3,4) for V3(R) with the standard inner product.

3.Let V be the set of all polynomial of degree min(m,n)d) r(A) k+1 in A is equal to

a) 0

b) 1

c) 2

d) 3

3.The system of equations written in the form Ax=B has a unique solution, if

a) r(A) = r(A,B)

b) r(A)< r(A,B)

c) r(A) = r(A,B) < no. of unknowns

d) r(A) = r(A,B) = no. of unknowns

4.The system of equations written in the form Ax=B has infinity of solutions, ifa) r(A) =(A,B)< no. of unknowns

b) r(A)=(A,B) = no. of unknowns

c) r(A) = r(A,B)

d) r(A) r(A,B)

5.The system of equations written in the form Ax = B has no solution, if

a) r(A)=r(A,B) b) r(A) (A,B) c) r(A)> r(A,B) d) r(A) + r(A,B)=0

6.If A and B are equivalent matrices, then

a) r(A) > r(B)

b) r(A) < r(B)

c) r(A) = r(B)

d) r(A) ( r(B)


Level : U

Unit : 5

Type : MC

Sub Unit : 5.4

1.The rank of is:a) 0

b) 1

c) 2

d) 3

2.The rank of is:a) 0

b) 1

c) 2

d) 3

3.If A is a matrix of the type 5 ( 3 and r(A) = 1, then every minor of order 2 is:

a) 0

b) 1

c) 2

d) 3

4.The system 3x + 4y = 7, 2x + y = 6 has

a) a unique solution

b) infinity of solutions

c) no solution

d) a solution provided x + y = 5

5.The system 2x + y = 7, x + 3y = 6 has

a) a unique solution

b) infinity of solutions

c) no solution

d) a solution satisfying the condition 2x + 3y = 5

6.For what value of a, does the system 3x + 4y = 7, 2x + ay = 5 have a unique solution

a) 3

b) 8

c) 3/8

d) 8/3


Level : K

Unit : 5

Type : VSA

Sub Unit : 5.4

1.Define rank of a matrix.

2.Mention any two of elementary transformation of a matrix.

3.Define equivalent matrices.

4.State the condition for the system of equations put in the form Ax = B to have a unique solution.

5.State the condition for the system of equations put in the form Ax = B to have infinity of solutions.

6.State the condition for the system of equations put in the form Ax = B to have no solution.


Level : U

Unit : 5

Type : VSA

Sub Unit : 5.4

1.What is the characteristic polynomial of the matrix?

2.What is the characteristic equation of the matrix?

3.What are the eigen values of the matrix ?

4.If A = , what is A2?


Level : U

Unit : 5

Type : PA

Sub Unit : 5.4

1.Find the rank of

2.Find the rank of

3.Show that the following equations 2x y + z = 7; 3x + y 5z = 13; x + y + z = 5 are consistent and solve them.

4.Examine for consistency the following system of equations and solve, if the system is consistent x + y + z = 6; -x + 2y - 4z = -9; 2x + 3y - 2z = 2.

5.Examine for consistency the following system of equations and solve, if the system is consistent 2x+3y-4z=4; 5x-2y+2z=-9; 5x+y-2z=-4.

6.Show that the equations x + y + 2z = 2; 2x y + 3z = 2; 5x y + Kz = 6 have a unique solution if k(8 and find all the solutions of the equations when k = 8.

7.Discuss for all values of a, the solution of the system of equations x + y + z = 2; 2x + y 2z = 2; ax + y + 4z = 2.

8.Prove that the following system is consistent and has a unique solution if a ( b, a(1, b(1, x + y + z = 6; x + 2y + 3z = 10; x + 2y + az = b.


Level : U

Unit : 5

Type : E

Sub Unit : 5.4

1.a)Show that the equations x + 2y = 3; y z = 2; x = yz = 1 are consistent.

b)Show that the equations

3x + 4y + 5z = a

4x + 4y + 6z = b

5x + 6y + 7z = c do not have a solution unless a + c = 2b.

2.a)Find the rank of .

(5 marks)

b)Find the values of a and b for which the equations

x + y + z = 3

x + 2y + 2z = 6

x + ay + 3z = b have

(i) a unique solution (ii) no solution

(10 marks)

3.a)Examine for consistency the following equations

2x + 6y + 22 = 0

6x + 20y 6z + 3= 0

6y 18z + 1 = 0

(5 marks)

b)Investigate the values of a and b for which the system of equations.

x + y + z = a

3x + 4y + 5z = b

2x + 3y + 4z = 1

(10 marks)

(i) a unique solution

(ii) infinity of solutions.

4.Diagonalise the matrix

5.Diagonalise the matrix

6.Find the characteristic equation of the matrix A = and hence evaluate A6-25A2+122A.

7.a)Find the eigen values of the matrix

b)If A =

; determine An in terms of A.

8.a)Verify Cayley-Hamiltons theorem for the matrix

b)If A =

, show that An = 7n-1 .

9.Find the Characteristic polynomial of the matrix A and hence compute 2A8-3A5+A4+A2-4I, Where A is

10. a)Find the eigen values of

b)If A = , determine An interms of A.


Level : U

Unit : 5

Type : MC

Sub Unit : 5.5

1.The eigen vales of are

a) 6,2

b) -6,2

c) 6,-2

d) 6 -2

2.The eigen vales of are

a) 4, -9

b) 2,3

c) 4,9

d) 2,9

3.The eigen values of are

a) 1,7

b) 2,3

c) 2,7

d) 0,7

4.If A = then A2 equals

a)

b)

c)

d)


Level : K

Unit : 5

Type : VSA

Sub Unit : 5.5

1.Define : characteristic polynomial of the square matrix A.

2.Define : Characteristic equation of the square matrix A.

3.Define eigen value of the matrix.

4.Define eigen vectors of the matrix.

5.What do you mean by diagnolisation of the matrix A.

6.Define similar matrices.

7.State Cayley Hamiltons theorem.


Level : U

Unit : 5

Type : PA

Sub Unit : 5.5

1.Find the eigen vales of

2.Find the eigen values of

3.Find the eigen values of

4.Find an eigen vector corresponding to one of the eigen values of

5.Calculate A4 when A = 1, by applying Cayley Hamiltons theorem

6.Find the characteristic equation of the matrix and hence obtain its inverse.

7.Characteristic equation of the matrix

8.Find the characteristic equation of the matrix and hence find its inverse.

9.Find An interms of the matrix A, where A =

10.If A = , show that An = 7n 1


Level : K

Unit : 5

Type : E

Sub Unit : 5.5

1.a)State and prove Caylay-Hamiltons theorem.

(9 marks)

b)If A = , evaluate A3 + A2

(6 marks)

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Algebra

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