Old Exam Papers June 2012 (Set 2)

8. (a) Let V be a finite dimensional vector space over a
field F and t: V V be a linear transformation.
If v1, v2 ,......, vn are distinct eigenvectors of t
corresponding to n distinct eigenvalues
1, 2 ,....., n respectively, then prove that
lv1 , v2 ,...., vnq is a linearly independent set.
(b) Prove that a n n square matrix A over a field F
is invertible if and only if det Abg 0 , and
1 1
detcA
h detbg
A
.
9. (a) If u and v are any two elements of a real inner
product space V, then prove that :
(i) u v u v
(ii) u v u v
(b) Let B lu 1, u2 ,....., unq be an orthonormal basis
of an inner product space V and v V be any
arbitrary vector. Then prove that the co-ordinates
of v relative to the basis B of V are v, ui ,
i 1, 2,......,
n and
800 4 MA/M.Sc.-MT-01
5. (a) Let K be a field extension of a field F and let
1, 2 ,......, n are elements in K which
are algebraic over F. Then prove that
F ba 1, a2 ,..... ang is a finite extension of F.
(b) Let F be a field, and let f xbg be a non-zero
polynomial in F x . Then prove that the splitting
field of f xbg is an algebraic extension of F.
6. (a) Let K be a normal extension of a field F and L is
an intermediate field, so that F L K , then
prove that K is also a normal extension of L.
(b) Let K be a Galois extension of a field F, then prove
that an element of K which remains invariant for
b g is
each member of the Galois group G K / F
necessarily a member of F.
7. (a) Let 2 2
t: R R be a linear transformation given
b gb g
by t x, y x y, x y , then find the matrix
of t with respect to basis B mb 1, 1gb , 1, 0gr.
(b) For any matrix A over a field F prove that rank
Abg= rank A T
of A .
c h where A T denotes transpose
MA/M.Sc.-MT-01 3 PTO