CLASS: M.Sc. MATHEMATICS
15N/455
ST. JOSEPH’S COLLEGE (AUTONOMOUS) TIRUCHIRAPPALLI – 620 002
SEMESTER EXAMINATIONS – NOVEMBER 2015
TIME: 40Minutes.
MAXIMUM MARKS: 30
SEM
SET
PAPER CODE
TITLE OF THE PAPER
I
2015
14PMA1104
LINEAR ALGEBRA
SECTION - A
Answer all the questions:
30  1 = 30
Choose the correct answer:
1.
2.
3.
4.
5.
6.
Any subset of a vector space which contains more than dim V
vectors is _______.
a) Linearly dependent
b) Linearly independent
c) Empty set
d) The space V
In a row-reduced matrix, each column which contains the leading
non-zero entry of some row has all its other entries
a) 1
b) 0
c) 2
d) n
Identify the non-elementary matrix from the following 2x2
matrices
1 𝑐
1 0
0 1
0 𝑐
a) [
]
b) [
]
c) [
]
d) [
]
0 1
0 𝑐
1 0
1 0
If dimW1 = 3, dimW2 = 5, dim(W1 ∩ W2) = 2 thendim(W1 + W2)=
a) 8
b) 10
c) 7
d) 6
A basis for a vector space V is a set of vectors which --a) Islinearly independent and spans V
b) Is linearly independent
c) spans V
d) Is linearly dependent and spans V
If A is an p x n matrix and B is an n x p matrix, then the matrix
BA is an ______ matrix.
a) n x n
b) m x m
c) m x n
d) n x m
7.
8.
9.
10.
11.
12.
13.
14.
15.
If the vector space V is finite dimensional then the nullity of linear
transformation T is the dimension of _______.
a) Range of T
b) V
c) Null space of T d) Empty set
If V and Ware vector spaces over the field F, a linear operator on
V is a linear transformation from -----------a) V into W
b) V into V
c) V into F
d) F into F
If F is a field and if T is an operator on F2 defined by
T(x1,x2) =(x1,0) and B = {e1 ,e2}is the standard basis then
[𝑇]𝐵 = ________.
0 1
0 0
0 0
1 0
a) [
]
b) [
]
c) [
]
d) [
]
0 0
1 0
0 1
0 0
In a vector space of dimension n , a subspace of dimension
______ is called a hyperspace
a) n +1
b) n – 1
c) n
d) 2 n
If V is a finite-dimensional vector space over the field F and W is
a subspace of V then dimV is _______.
a) dim V0 + dim W
b) dim V0 + dim W0
c) dim W - dim W0
d) dim W + dim W0
The null space of the transpose of a linear transformation T from
vector space V into vector space W is the annihilator of the _____
of T.
a) range of T
b) transpose
c) rank of T
d) null space
If f, g and h are polynomials over the field F such that f≠0 and
fg = fh then _______.
a) f = h
b) f = g
c) g = h
d) f = g = h
If f is a polynomial over the field F and c is an element of F then f
is divisible by x-c if and only if
a) f(c) = 1
b) f(c) = 0
c) f(c) ≠ 0
d) f(c) ≠ 1
If c is a root of the polynomial f, then the multiplicity of c as a
root of f is the largest positive integer r such that ______ divides f.
a) (x+c)r
b) (x-c)r
c) (x+c)r+1
d) (x-c)r-1
16. The polynomial f in F[x] is said to be _______ if there exists no
polynomials g, h in F[x] of deg ≥ 1 such that f = gh.
a) reducible
b) irreducible
c) prime
d) monic
17. The polynomial x2 +1 is _______ over the field R of real numbers.
a) irreducible b) reducible
c) monic
d) scalar
18. The field F is algebraically closed if every prime polynomial over
F has degree _______.
a) 1
b) 0
c) > 1
d) any finite positive integer
19. If V is a vector space over the field F and T is a linear operator on
V then any vector α such that Tα=cα is called a _______ of T
associated with the characteristic value c.
a) latent root
b) characteristic vector
c) spectral value
d) eigen value
20. If T is a linear operator on a finite-dimensional space V and c is a
characteristic value of T then ________.
a) det(cT-I) = 0
b) det(T-cI) = 0
c) det(T-I) = 0
d) det(T-cI) = 1
21. If A is an n×n matrix over the field F , a characteristic value of A
in F is a scalar c in F such that the matrix (A-cI) is _______.
a) invertible b) nonsingular
c) singular
d) none of these
22. If A=[0 −1]represents a linear operator T on 𝑅2 in the standard
1 0
ordered basis then the characteristic values of T are _______.
a) 1 and -1
b) i and –i
c) 1 and 0 .5 d) no
23. If T is a linear operator on the finite dimensional space V then T is
_______ if there is a basis for V, each vector of which is a
characteristic vector of T.
a) diagonalizable
b) T-Conductor
c) triangulable
d) none
24. If D is any alternating 𝑛 – Linear function on 𝐾 𝑛 𝑋 𝑛 , then for
each 𝑛 𝑋 𝑛 matrix 𝐴
b) D(A) = det D(I)
a) D(A) = D(I)
d) None of these
c) D(A) = det A
25. If W is a subspace of a vector space V and if T is a linear operator
on V then W is _____ under T if for each vector α in W the vector
Tα is in W .
a) null space b) T-conductor
c) invariant
d) annihilator
26. If W is an invariant subspace for T and α be a vector in V then ---_______ of α into W is the set of all polynomials g such that g(T)α
is in W.
a) T-annihilate b) null space
c) invariant
d) T-conductor
27. If W is an invariant subspace for the linear operator T, the
conductor S(α ; W) is _______ in the polynomial algebra F[x].
b) a T-conductor
a) an ideal
d) invariant
c) a T-annihilate
28. The linear operator T is _______ if there is an ordered basis in
which T is represented by a triangular matrix
b) invariant
a) triangulable
d) annihilator
c) diagonalizable
29. A projection of vector space V is a linear operator E on V such
that _______.
a) E=1
b) E=0
c) E2 = E
d) E2 =1
30. If the minimal polynomial for a linear operator T on the finite
dimensional vector space V over F decomposes over F into a
product of linear polynomials then there is a diagonalizable
operator D on V and a nilpotent operator N on V such that _____.
b) T=D-N & DN = ND
a) T=D+N & DN = ND
d) T=D/N & DN = ND
c) T=DN & DN = ND
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