MATHEMATICS MODEL PAPER FIFTH SEMESTER PAPER 6 – LINEAR ALGEBRA COMMON FOR B.A & B.Sc (w.e.f. 201516 admitted batch)
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MATHEMATICS MODEL PAPER FIFTH SEMESTER
PAPER 6 – LINEAR ALGEBRA COMMON FOR B.A & B.Sc
(w.e.f. 201516 admitted batch)
SECTION
– A


I.

Answer
any five questions. Each question carries five marks.

5
x 5 = 25 M


1.

Let p, q, r be the fixed elements of a
field F. Show that the set W of all triads (x, y, z) of elements of F, such
that px + qy + rz = 0 is a vector subspace of V_{3}(F).


2.

Express the vector a = (1, –2, 5) as a linear combination of the vectors e_{1}
= (1, 1, 1), e_{2} = (1, 2, 3) and e_{3} = (2, –1, 1)


3.

If a, b, g
are L.I vectors of the vector space V(R) then show that a + b,
b + g, g
+ a are
also L. I vectors


4

Describe explicitly
the linear transformation T : R^{2} ® R^{2}
such that T (1, 2) = 3,0) and T(2,1) = (1, 2)


5.

Let U(F) and V(F0 be two vector space and T : U(F) ® V(F) be a linear
transformation. Prove that the range set R(T) is a substance of V(F).


6.

Solve the system 2x
– 3y + z = 0, x + 2y – 3z = 0, 4x – y – 2z = 0


7.

State and prove
Schwarz inequality.


8.

is an orthogonal set
of the inner product space R^{3}®


SECTION – B


II.

Answer
the all five questions. Each carries ten marks

5
x 10 = 50 M


9.

a)

Prove that the subspace W to be a subspace
of V(F) Û aa + bb
Î W "a, b Î F and a, bÎ W.



(OR)



b)

Prove that the four vectors a = (1, 0, 0), b
= (0, 1, 0), g = (0, 0, 1), d =(1, 1, 1) in
V3(C) form a linear dependent set, but any three of them are linear
independent.


10.

a)

= dim (V) – dim (W).



(OR)



b)

Let W_{1} and W_{2} be two
subspaces of a finite dimensional vector space V(F). Then prove that dim
(W_{1} + W_{2}) = dim (W_{1})
+ dim (W_{2}) – dim (W_{1} Ç W_{2})


11.

a)

State and prove RankNullity theorem.



(OR)



b)

Define a linear transformation. Show that
the mapping T : R^{3} ® R^{2}
is defined by T (x, y, z) = (x – y, x –z) is a linear transformation


12.

a)

State and prove Cayley –Hamilton theorem.



(OR)



b)

Find the characteristic roots and the
corresponding characteristic vectors of the matrix


13.

a)

State and prove Bessel’s inequality.



(OR)



b)

Applying GramSchmidt orthogonalization
process to obtain an orthonormal basis of R^{3}(R) from the basis


THE
END