Download SGBAU (Sant Gadge Baba Amravati university) BSc 2019 Summer (Bachelor of Science) 6th Sem Mathematics Linear Algebra Previous Question Paper
B.Sc. Part-III (Semester?Vl) Examination
MATHEMATICS
(Linear Algebra)
Paper?XI
Time : Three Hours] [Maximum Marks : 60
Note :?-(1) Question No. l is compulsory and attempt this question once only.
(2) Attempt ONE question from each unit.
, 1. Cheese the correct altcmative (1 mark each) :
'(i) S is a non?empty subset of vector space V, then the smallest subspace of V containing
S is :
(a) S (b) {S}
(c) [S] ((1) None
(ii) Let U and V be ?nite dimensional vector spaces and T : U ?> V be a linear map one-
one and onto, then :
(a) dim U = dim V (b) dim U at dim V
(c) U = V (d) U ? d
(iii) Let W is subspace of vector space V. Then {fe V/f(w)=0, ? w e W} is called as :
(a) [Iilatory of W (b) Annihilator of W
(c) Dual space of W (d) None
(iv) The normalized vcctor (l, ?2, 5) is :
, , l ? 2 .5
(a) (1, *2, D) (b) (?r?go?j]
1 5 1 ?2
(c) [3'15] (d) [???J
(v) In IPS V(F) the relation H u + v ?2 + H u ? v I!2 = 2 (1| u H2 + H v H?) is called as :
(a) Schwartz inequality (b) Triangle law
(c) Parallelogram law (d) Bcssels inequality
(Vi) For two subspaces U and W of V(F). V = U ? W m ...............
(a)UmW={0} (b)V=U+W
(c) U n W = {0} and V = U + W ((1) None of these
YBC?l 5328 l ? (C0nld.)
(vii) Let T : M ?> N be an R-iomomorphism. If B is a submodule (if N, then :
(a) T"(B) is submodulc of N (b) T 1(B) is submoc ule of M
(c) T?(B) is kernel of R--homamorphism (d) T?(B) = T(M)
(viii)[f T : U -?> V then the set lTiu) I u e L'} ?? ...........
(a) Ker F (17) NW
(c) R(T?: id) None 01' these
(ix) If [i V li = 1, then V is called :
(a) Normalised (b) Orthonormal
(_c) Scalar inner product (d) Standard inner product
(x) If V is n-dimensional, then the dimension of V is :
(a) Less than n (b) Greater than n
(c) Equal n (d) Zero 10
UNIT?l
2. (a) Let U and W be two subspaces of a vector space V and Z = U + W. Then prove
that Z = U 63 W iff 2: e Z, x "- u ~ w is unique represeitation for u e U and
w e W. 5
(b) Extend the linearly indepcrdcnt set {(1. 1, 1. 1) (1. 2. l, 2)} in V4 to a basis for
V4. 5
3i (p) If U and W are ?nite dimensional subspaces of vector space V, then prove that :
dim(U + W) = dim 1,7 + dim W ?? dim(U n W). 5
(q) Let R? be the set of all positive real numbers. De?ne the opeiations of addition 63 and
scalar multiplication <5: :15 r?nllmxs :
u63v=u-\'Vu,\'e R?
andoc?u=u?,>o?uc R?andace R
Prove that R+ is a real vector space. 5
UNlT?ll
4. (a) If U, V is a vector spice over a ?eld F and T : U ?> V be a linear, then prove that :
T(UHUl + (quz + ...... + Gnu") = (1' DUI) v uszz) + ....... 1 anT(u?)
VuleUJxIeRISiSnandneN.
2
(b) Let 'I' : V4 ??> V3 be a linear map de?ned by T(el) ?~ (1, 1, l), T(ez) -??' (l, ~l, l),
T(eJT? = (l. 0. 0), T(Q) : (I. (J, 1?).
Verify Rank-nullity :heorem. 4
YBC?15328 2
(Comd.)
(C)
5- (D)
(q)
6. (a)
(b)
7. (p)
(q)
(r)
8 (a)
(b)
(C)
9- (1?
Find the matrix of the linear map T ; v2 ?> V; de?ned by T(x, y) = (?x + 2y, y, -?3x + By)
related to the bases Bl = {(l, 2), (?2, 1)} and B2 = ((?l, 0, 2), (1, 2, 3), (l, *1, 1)}.
4
Let U and V be vector spaces over the same ?eld F. Then prove that
function T : U -> V is linear iff T(au + pv) = aT(u) + [5T(v), v a, [3 e F and
u, v e U. 5
If matrix of a linear map T with respect to bases BI and B2 is :
?l 2 l
l 0 3
where BI = {(1, 2, O), (0, ~l, 0), (l, ?l, 1)} and B? = {(1, 0), (2, ?l)}, then ?nd
T(x, y, 2)- 5
UNlT?III
Let V be the space of all real valued continuous functions of real variable. De?ne
T : V ?> V by
(Tom: jmdt, v fe v, x e R.
0
Show that T has no eigen value. 5
Prove that if V be a ?nite dimensional vector space over F and v(?0) e V, then
3 f e \?I such that f(v) :2 o. 5
If WI and W2 are subspaces of a ?nite dimensional vector space V, show that
A(wl + W) = A(Wl) n A(WZ). 5
If K1 is eigenspace. then prove that K,t is a subspace of vector space V. 3
Define characteristic root and characteristic vector. 2
UNIT?IV
[n F"n de?ne for u=(al,a2,a3, ------ ,0") and V=(Blyl32 ------ B?)
(u, 0:05.61 +0562 4- .......... + aan'
Show that this de?nes an inner product. 4
If {xv x2, x3, ....... xn} be an orthogonal set, then prove that :
[|xl+x2+x3+ .......... +xn||2=||xl||2+||x2||2+ ........... +l|xn||2 4
Prove that orthogonal complement i.e. W? is subspace of V.
m
[f {w1, w2, ....... wm} is an orthonormal set in V, then 2 |(wi,v)|2 s||vu2 for any
i=l
v e V. 4
YBC?15328 3 (Contd.)
(q)
(r)
10. (a)
(b)
(C)
11- (p)
(q)
If V is a ?nite dimensional inner product space and W is a subspace of V then prove
that (Wl)* = W. 4
(i) De?ne inner product in vector space. 1
(ii) De?ne orthogonal set. 1
UNlT?V
Let A be a submodule ot'an R-module M and T is a mapping from M into M/A de?ned
by Tm = A + m, v m a M Then prove that T is an R-homomorphism of M into
WA and ker T = A. 5
Let T bx: a homomorphism of an R-module M to an R-module H. Prove that 'l' is one-
one iff ker T = {0}. 3
De?ne :
(i) Submodulc
(ii) Unital R-module. 2
If A and B are submodules of M. then prove that 5:2 is isomorphic to
E13? 6
Prove that arbitrary infersection of suhmodulcs of a module is a suhmodule. 4
YBCAISJZS 4 525
This post was last modified on 10 February 2020