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

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B. TECH.

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THEORY EXAMINATION (SEM–VIII) 2016-17

APPLIED LINEAR ALGEBRA

Time: 3 Hours

Max. Marks : 100

Note: Be precise in your answer. In case of numerical problem assume data wherever not provided.

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SECTION – A

1. Attempt all parts of the following questions: 10 x 2 = 20
   1. Find dimension of vector space C(R).
   2. Define Basis of a vector space.
   3. State rank-nullity theorem.
   4. Let T: R² ? R² be a linear transformation such that T = = What is the value of T value

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   5. Find all non-singular linear transformation T : R⁴ ? R³.
   6. Find the condition that T is non-singular.
   7. Define complete ortho normal set.
   8. Give polarization identity.
   9. A real quadratic form in three variables is equivalent to the diagonal form 6y₁² +3y₂²+0y₃² then find the quadratic.

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   10. Define linear functionals with examples

SECTION - B

2. Attempt any five parts of the following questions: 5 x 10 = 50
   1. Define field with example
   2. Prove that the set :a,b ? R is vector space over R.

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   3. Find the Eigen values and Eigen vectors of the matrix A=
   4. The matrix of quadratic form qon R³ given by q(x₁,x₂, x₃) = x₁² - x₃² + 3x₁x₂-6x₂x₃
   5. State and prove Minkowski inequality.
   6. Let T be the linear transformation on V such that T³ -T² -T + I = 0, then find T^-1.
   7. Let V be a finite dimensional inner product space and S, S₁, S₂ are subset of V Prove that (i) S⁺ = {S} (ii) {S} = S⁺⁺

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   8. Prove that union of two subspaces is subspace if one is contained in other.

SECTION – C

3. Attempt any two parts of the following questions: 2 x 15 = 30
   1. Prove that the system of three vectors (1,3,2), (1, -7, -8), (2,1,-1) of V₃(R)is linearly dependent.
   2. Let T : R² ? R³ be a linear transformation given by T(x₁, x₂) = (X₁ + X₂, X₁ - X₂, X₂) then find the rank of T.

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4. 1. W = Span{x₁, x₂}, where x₁ = x₂ = Construct orthogonal basis (v₁, v₂) for
   2. Let A = and v₁ ,v₂ Eigen vectors of A
5. 1. Define a linear transformation T : R² ? R by T(x) = Find the images under of u = and u + v =

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   2. Find a vector_x=(c,d) that has dot product x.r=1 and x.s=0 with the given vectors r = (-2,1), s = (-1,2)

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