Download AKTU B-Tech 8th Sem 2016-17 Applied Linear Algebra Question Paper

Download AKTU (Dr. A.P.J. Abdul Kalam Technical University (AKTU), formerly Uttar Pradesh Technical University (UPTU) B-Tech 8th Semester (Eight Semester) 2016-17 Applied Linear Algebra Question Paper

PrintedPages :2 Roll No.| l l l l I l l l l l NOE049
B. TECH.
THEORY EXAMINATION (SEM?VIII) 2016-17
APPLIED LINEAR ALGEBRA
Time : 3 Hours Max. Marks : 100
Note : Be precise in your answer. In case Ofnumerical problem assume data wherever not provided.
SECTION ? A
1. Attempt all parts of the following questions: 10 x 2 = 20
a) Find dimension of vector space C (R) .
b) Define Basis of a vector space.
c) State rank-nullity theorem.
1 1
(1) Let T : R2 ?> R2 be a linear transformation such that TKOJ] =[ j and
Wamwm
e) Find all non-singular linear transformation T : R4 ?> R3 .
f) Find the condition that T is non-singular.
g) Define complete ortho normal set.
h) Give polarization identity.
i) A real quadratic form in three variables is equivalent to the diagonal form
63112 + 3 yz2 +03232 then find the quadratic form.
j) Define linear functionals with examples
SECTION ? B
2. Attempt any five parts of the following questions: 5 x 10 = 50
a) Define field with example.
a a + b .
b) Prove that the set V = :a,b e R 1s vector space over R .
a + b b
c) Find the Eigen values and Eigen vectors of the matrix
3 1 4
A = 0 2 6
0 0 5
d) The matrix of quadratic form q on R3 given by q(x1,x2, x3) = x12 ? x32 +3x1x2 ? 6x2x3
e) State and prove Minkowski inequality.
f) Let T be the linear transformation on V such that T3 ? T2 ?T+ I = 0, then find T_l .
g) Let V be a finite dimensional inner product space and S , 51,52 are subset of V Prove
that (1) SL = {s}T (ii) {s}=sii
h) Prove that union of two subspaces is subspace if one is contained in other.

SECTION ? C
Attempt any two parts of the following questions: 2 x 15 = 30
3.
(i)
(ii)
(i)
(ii)
(i)
(ii)
Prove that the system of three vectors (1,3, 2), (1, ?7,?8), (2,1,?1) of V3(R) is
linearly dependent.
Let T : R2 ?) R3 be a linear transformation given by T(x1,x2) = (x1 +x2, x1 ? x2,x2)
, then find the rank of T .
3 1
W = Span {x1,x2} , where x1 = 6 ,x2 = 2 .Construct orthogonal basis (121,122) for
0 2
1 6 6 3 .
Let A = ,u = ,v .Are u and v Elgen vectors of A.
5 2 ?5 ?2
0 ?1 x ?x
Define a linear transformation T : R2 ?> R2 by T(x)=[1 ][ 1]=[ 2
0 x2
. 4 2 6
the1magesunderTofu= 1 ,v 3 and u+v= 4
Find a vector x: (ad) that has dot product x.r =1and x.s = Owith the given
vectors r = (?2, 1), s = (?1, 2)
:| . Find
X1