VIVEKANANDHA COLLEGE OF ARTS AND SCIENCES FOR WOMEN

(Autonomous)
Elayampalayam, Tiruchengode – 637 205. Namakkal.

DEPARTMENT OF MATHEMATICS

Question Bank – 2023-24 (Even Sem)

Programme(s): B.Sc Subject Code: 21U6MAC13

Subject Name: Linear algebra 21U6MAC13

Section - A
Let V be a non empty set over a field F, together with binary operation (+,•) is called Unit-1 K2 CO-1
1
as___
a. Vector space b. Dual space
c. Quotient space d. Metric space
2 If every α∈F, v∈V α(u+v) =___ Unit-1 K2 CO-1

a. αu-αv b. αuv
c. αu+αv d. αvu
___is not a vector space over R. Unit-1 K1 CO-1
3
a. R²×R b. R×R
c. R³×R d. None of these
4 The intersection of any two subspaces of vector space is to subspaces of a___ Unit-1 K2 CO-1

a. Equal b. Need Not equal

c. Same d. Not same
5 ___ of the two subspace is a subspace if and only if one is contained in the other. Unit-1 K1 CO-1

a. Intersection b. Difference
c. Complement d. Union
6 A homomorphism T of a vector space is also called ___ transformation Unit-1 K2 CO-1

a. Bilinear b. Unlinear
c. Linear d. None of these
Let T: U->V be a homomorphism then the kernel of T is said to be Ker (T) = {u∈U /uT Unit-1 K3 CO-1
7
=__}
a. 0 b. 1
c. Infinity d. All of these
8 The kernel of a homomorphism is a ___ Unit-1 K3 CO-1

a. Vector space b. Quotient space

c. Subspace d. None of these
9 The intersection of an arbitrary collection of subspace of a vector space is the___ Unit-1 K2 CO-1

a. Subspace b. Finite
c. Infinite d. None of these
Unit-1 K2 CO-1
10 Let V be a vector space over a field F, let A and B be subspace of V then A+B/B 〰️___

a. A/AUB b. A/A+B
c. A+B/A d. A/AnB
Section - B
1 The intersection of any two subspaces of vector space is to subspace of a same. Unit-1 K2 CO-1
2 R×R is not a vector space over R. If we define addition and multiplication as (a,b) + Unit-1 K2 CO-1
(c,d) = (a+c,b+d)
α(a,b) = (αa,b) ∀ α∈R

3 The kernel of a homomorphism is a subspace. Unit-1 K2 CO-1

4 Show that any field k is a vector space over any of its subfield F. Unit-1 K1 CO-1
5 If A and B are the subspace of V then prove that A+B ={v ∈ V/v = a+b; a∈A, b∈B } is a Unit-1 K1 CO-1
subspace of V further A ≠ B is smallest subspace containing A and B, if any subspace
of V containing A and B then w contain A ≠B

Section - C
1 State and prove that Fundamental Theorem of Homomorphism Unit-1 K1 CO-1
2 Union of the two subspaces is a sub space iff one is contained in the others. Unit-1 K2 CO-1
3 Show that quotient space is a vector space over F. Unit-1 K1 CO-1
4 Let R be a vector space over a field F then, Unit-1 K3 CO-1
I) α.0 = 0 ∀ α∈F
II) 0.v = 0 ∀ v∈R
III) (-α) v = -(αv)-α(-v) ∀ α∈F, v∈R
IV) α (v1-v2) = (αv1-αv2) ∀ α∈F, v1,v2∈R
V) αr = 0 α = 0 or r = 0
VI) αv1 = αv2 α ≠ 0, v1=V2

Section - A
1 L(S) is a____of v Unit-2 K2 CO-2
a. Subspace b. Subset
c. Linear span d. None of these
2 L(S) is called the ____by the set S Unit-2 K1 CO-2
a. Subset spanned b. Subspace spanned
c. Spanned d. None of these
3 L(L(S))=L(S) Unit-2 K2 CO-2
a. TRUE b. FALSE
c. Both A and B d. None
4 L(SUT)=L(S)+L(T) Unit-2 K2 CO-2
a. TRUE b. FALSE
c. Both A and B d. None
5 Any subset of a linearly independent set is____. Unit-2 K2 CO-2
a. Dependent b. Independent
c. Linearly independent d. Linearly dependent
6 A subset os a vector space V is called ____. Unit-2 K1 CO-2
a. Basis b. Subset
c. Subspace d. none of these
7 If V is finite dimensional over F then any two bases of V have the ____number element. Unit-2 K2 CO-2
a. One b. Different
c. Same d. Zero
8 dim V/W=dim V+dim W Unit-2 K1 CO-2
a. TRUE b. FALSE
c. Both d. None
9 Any two ____dimensional vector space over F of the same dimension are isomorphic. Unit-2 K1 CO-2
a. Infinite b. Finite
c. Different d. None of these
10 dim (A+B) =dimA+dimB Unit-2 K1 CO-2
 Unit-2 K1 CO-2
a. TRUE b. FALSE
c. Both d. None
Section - B
1 Prove that L(S) is a subspace of V. Unit-2 K1 CO-2
2 If v1,v2,..., vn ∈V are linearly independent, then every element in their linear span has a Unit-2 K2 CO-2
unique representation in the form λ1v1+λ2v2+..... +λn vn with the λi∈F.
3 If V is finite-dimensional over F then any two bases of V have the same number of Unit-2 K3 CO-2
elements.
4 F^(n) is isomorphic F^(m) if and only if n = m. Unit-2 K2 CO-2
5 If A and B are finite- dimensional subspaces of a vector space V, then A + B is finite- Unit-2 K4 CO-2
dimensional and dim (A + B) = dim (A) + dim (B) -dim (A n B).
Section - C
1 Unit-2 K3 CO-2
Let V be a vector space over a field F, let S & T be the subsets of V then,
1. S c T implies L(S) c L( T).
2. L(S u T) = L(S) + L( T).
3.L(S)=S iff S is the subspace of V.
4. L(L(S)) = L(S).

2 If V is finite-dimensional and if W is a subspace of V, then W is finite-dimensional, dim Unit-2 K2 CO-2

W ≤ dim V and dim V/W = dim V - dim W.
3 If v1,v2,.....,vn in V have W as linear span and if v1,v2,....,vk arelinearly independent, Unit-2 K4 CO-2
then we can find a subset of v1, v2,...,vn of the form v1, v2,.....,vk,vi1,.....,vir consisting
of linearly independent element whose linear span is also W.
4 Unit-2 K2 CO-2
If v1, v2,...,vn is a linear span of V over F & if w1, w2,...,wn in V are linearly
independent over F, then m≤n.

Section - A
1 ||v||=_____ Unit-3 K2 CO-3
a. (v,v) b. (v,v)^1/2
c. (v,v)^² d. None
2 U is orthogonal to V if______ Unit-3 K2 CO-3
a. (U,V)=0 b. (U,V)=0
c. both a & b d. None
3 {vi} is an orthonormal set if______ Unit-3 K3 CO-3
a. ength of {vi} is 0 b. length of {vi} is 1
c. length of (vi,vj) is 0 d. both b & c
4 Inner product space is a_____ Unit - 3 K1 CO-3
a. Linear space b. Banach space
c. Hilbert space d. Vector space
5 An inner product space v of complete field c is called an_____ Unit-3 K1 CO-3
a. Unitary space b. not unitary space
c. Vector space d. All the above
6 If V is a inner product space then _____ Unit-3 K3 CO-3
a. ||v||=√<v,v> b. ||v||*√<v,v>
c. ||v||+√<v,v> d. None
7 If v is an inner product space for any two vector u,v∈V then _____ Unit-3 K4 CO-3
a. ||u+v||≤||u||+||v|| b. ||u+v||<||u||+||v||
c. ||u+v||≥||u||+||v|| d. All the above
8 An inner product space is called an ______ according as F is the field of real numbers. Unit-3 K2 CO-3
a. Space b. Euclidean space
c. Unitary space d. Vector space
9 Every finite dimensional inner product space has _______ basis. Unit-3 K3 CO-3
a. Orthogonal b. Orthonormal
c. Both a&b d. None
10 ||u+v||2 + ||u-v||2=_____ Unit-3 K4 CO-3
a. 2[||u||^²+||v||^²] b. both a&b
c. 2[||u||^2-||v||^²] d. None
Section - B
1 State and prove parallelogram law in vector space v. Unit-3 K1 CO-3
2 State and prove triangular inequality. Unit-3 K1 CO-3
3 Let v be the set of all continuous complex valued function on the closed unit interval Unit-3 K3 CO-3
[0,1]if f(t), g(t)V, Defined [f(t), g(t)]= ∫_0^1▒𝑓(𝑡)
(𝑔(𝑡)) ̅ dt
4 Orthogonal complement is a subspace of v. Unit-3 K2 CO-3
5 If vi is orthonormal then the vector in {vi} are linealy independent. Unit-3 K2 CO-3
Section - C
1 State and prove Schwarz inequality Unit-3 K1 CO-3
2 If F be a real field and let V be the set of polynomial in a variable x over F of degree less Unit-3 K3 CO-3
than or equal to two.In this we define an inner product space by if p(x),q(x) V then
<p(x),(x)>=
∫_−1^1▒𝑝(𝑥)𝑞(
𝑥) 𝑑𝑡

3 State and prove Gram Schmidt Orthogonalisation Process Unit-3 K1 CO-3

4 If V is finite dimensional inner product space.If w is a subspace of v.Then v=u+w Unit-3 K4 CO-3
particularly v is the direct sum of w and w⊥ .

Section - A
1 An element in A(V) which is not regular is called ____ Unit-4 K3 CO-4
a. Regular b. Singular
c. Invertible d. None
2 If T is both right and left invertible and if TS=ST=____ Unit-4 K3 CO-4
a. 0 b. 2
c. 1 d. 3
3 If V is finite dimensional over F for ST belongs to A(V) is Unit-4 K2 CO-4
a. r(ST)=r(T) b. r(ST)<r(T)
c. r(ST) ≤r(T) d. r(ST)≥r(T)
4 The element v≠0 in v is called a______ Unit-4 K1 CO-4
a. Characteristic vector b. Eigen vector
c. Characteristic roots d. Both A and B
5 The range of T is all of V iff T is _____ Unit-4 K2 CO-4
a. One to one b. Bijection
c. Onto d. Range
6 A homomorphism T of vector space is called a _____transformation. Unit-4 K1 CO-4
a. Linear b. Translation
c. Bilinear d. Rotation
7 An element in A(V) which is not regular is called ______. Unit-4 K1 CO-4
a. Singular b. Finite
c. Infinite d. None of these
8 T∈ A(V) is right invertible then it is _____. Unit-4 K1 CO-4
a. Invertible b. Singular
c. Regular d. None of these
9 T∈ A(V) is not singular then T ∈ A(V) is _____. Unit-4 K2 CO-4
 Unit-4 K2 CO-4
a. Regular b. Not regular
c. Finite d. None of these
10 Any set of linearly independent vectors in V can have atmost _____ elements. Unit-4 K4 CO-4
a. 2 b. 3
c. 4 d. n
Section - B
1 Unit-4 K2 CO-4
Let A be an algebra with unit element over F and suppose that A is dimension 'm' over
F.Then every element in A satisfies some non trivial polynomial in F(x) of degree almost
m.
2 State Cayley Hamilton theorem Unit-4 K1 CO-4
3 Consider the matrix. Unit-4 K4 CO-4
[ 5 -6 -6
-1 4 2
3 -6 -4]
Find the minimal polynomial.

4 If V is finite dimensional over F the T ∈ A(v) regular iff T maps V onto v. Unit-4 K1 CO-4
5 If V is finite dimensional over F then T ∈ A(v) singular off there exist a V not equal to 0 Unit-4 K3 CO-4
in V such that VT=0.
Section - C
1 If A is an algebra with unit element over F, then A is isomorphic to a subalgebra of A(v) Unit-4 K2 CO-4
for some vector space over F.
ii) r(TS) ≤ r(T) ( and so r(ST) < min {r(T),r(S)}
2 iii) r(ST) = r(TS) = r(T) for S regular in A(v) Unit-4 K4 CO-4
3 If λ1, λ2,......λ k in F are distinct char root of T ∈ A(v) and if V1,V2,...,Vk are char vector Unit-4 K3 CO-4
of T belonging to λ1, λ2,......λ k respectively then V1,V2,...Vk are linearly independent
over F.
4 Hom (v,v) or A(v) is an algebra over F. Unit-4 K2 CO-4

Section - A
1 An element T belongs to A(V) is called ___ Unit-5 K1 CO-5
a. Nilpotent b. Nilpotent Transformation
c. Both a &b d. None
2 If T belongs to A(V) is nilpotent ,then K is called the __ Unit-5 K1 CO-5
a. Nilpotent b. Nilpotent transformation
c. Index of nilpotent d. None
3 There exists a subspace W of V is invariant under T then __ Unit-5 K2 CO-5
a. V=V1+W b. V1 n w1
c. Both a&b d. None
4 The integer n1,n2,.....nr are called the invariant of ___ Unit - 5 K2 CO-5
a. K b. T
c. L d. A
5 If cyclic rule is__ Unit-5 K2 CO-5
a. MT^m= (0) b. MT^m ≠ (0)
c. An element Z ∈M d. All of these
6 Two nilpotent linear transformation are similar iff they have the __ Unit-5 K2 CO-5
a. Different invariants b. Same invariants
c. Both a&b d. None
7 If S and T be the two nilpotent is called __ Unit-5 K1 CO-5
a. Linear transformation b. Linear combination
c. All d. None
8 If M, of dimension M ,is __ Unit-5 K2 CO-5
 Unit-5 K2 CO-5
a. Cyclic b. Nilpotent
c. Dimension d. All of these
9 If V1,V2,........ Vr are __ Unit-5 K2 CO -5
a. Integers b. Cyclic
c. Invariant d. None
10 If the series VT^n1-k + VT^n1-k+1 ,..........VT^n1-1 are __ Unit-5 K2 CO-5
a. Linearly dependent b. Linearly independent
c. Linearly combination d. Linearly transformation
Section - B
1 If T ∈ A(V) is Nilpotent, Then α0 +α0 T+......+ αm T^m where αi ∈ F, is the invertible if Unit-5 K2 CO-5
α0≠ 0.

2 If u ∈ v1, is such that UT^h1-k = 0, where 0<k≤n, then u=u0 T^k for some u0 ∉ v1. Unit-5 K3 CO-5
3 If M of dimension M is cyclic with respect to T then the dimension of MT^k is M-K-V K ≤ Unit-5 K4 CO-5
M
4 For A,B ∈ Fn & λ∈ F Unit-5 K2 CO-5
i) tr(λA) = λ tr A
ii) tr (A+B) = tr A +tr B
iii) tr (AB) = tr(BA)

5 If T is nilpotent, then tr T^k = O , K=1,2,3.......n. Unit-5 K3 CO-5

Section - C
1 If T ∈ A(v) is nilpotent, then α0+α1+.......+αa m T^m where αi ∈ F, is the invertible if α0 Unit-5 K2 CO-5
≠ 0.
2 Two nilpotent linear transformation are similar if they have thee same invariants. Unit-5 K4 CO-5
3 If T ∈ A(v) then trT is the sum of the characteristic roots of T ( using each characteristic Unit-5 K3 CO-5
roots as often as it multiplicity ).
4 Prove that Unit-5 K1 CO-5
1) (A')'=A
2) (A+B)'=A'+B'
3) (AB)'=B'A'