MATHEMATICS MODEL PAPER FIFTH SEMESTER PAPER 6 – LINEAR ALGEBRA COMMON FOR B.A & B.Sc (w.e.f. 2015-16 admitted batch)

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MATHEMATICS MODEL PAPER FIFTH SEMESTER
PAPER 6 – LINEAR ALGEBRA COMMON FOR B.A & B.Sc
(w.e.f. 2015-16 admitted batch)
SECTION
– A
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I.
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Answer
any five questions. Each question carries five marks.
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5
x 5 = 25 M
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1.
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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 V3(F).
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2.
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Express the vector a = (1, –2, 5) as a linear combination of the vectors e1
= (1, 1, 1), e2 = (1, 2, 3) and e3 = (2, –1, 1)
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3.
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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
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4
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Describe explicitly
the linear transformation T : R2 ® R2
such that T (1, 2) = 3,0) and T(2,1) = (1, 2)
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5.
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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).
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6.
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Solve the system 2x
– 3y + z = 0, x + 2y – 3z = 0, 4x – y – 2z = 0
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7.
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State and prove
Schwarz inequality.
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8.
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is an orthogonal set
of the inner product space R3®
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SECTION – B
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II.
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Answer
the all five questions. Each carries ten marks
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5
x 10 = 50 M
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9.
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a)
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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.
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(OR)
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b)
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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.
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10.
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a)
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= dim (V) – dim (W).
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(OR)
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b)
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Let W1 and W2 be two
subspaces of a finite dimensional vector space V(F). Then prove that dim
(W1 + W2) = dim (W1)
+ dim (W2) – dim (W1 Ç W2)
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11.
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a)
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State and prove Rank-Nullity theorem.
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(OR)
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b)
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Define a linear transformation. Show that
the mapping T : R3 ® R2
is defined by T (x, y, z) = (x – y, x –z) is a linear transformation
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12.
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a)
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State and prove Cayley –Hamilton theorem.
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(OR)
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b)
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Find the characteristic roots and the
corresponding characteristic vectors of the matrix
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13.
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a)
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State and prove Bessel’s inequality.
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(OR)
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b)
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Applying Gram-Schmidt orthogonalization
process to obtain an orthonormal basis of R3(R) from the basis
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THE
END