Download SGBAU BSc 2019 Summer 6th Sem Mathematics Linear Algebra Question Paper

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B.Sc. Part-II (Semester-VI) Examination

MATHEMATICS

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(Linear Algebra)

Paper—XI

Time : Three Hours] [Maximum Marks : 60

Note :— (1) Question No. 1 is compulsory and attempt this question once only.

(2) Attempt ONE question from each unit.

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1. Choose the correct alternative (1 mark each) :

(i) S is a non-empty subset of vector space V, then the smallest subspace of V containing S is:

(a) S (b) {S}

(c) [S] (d) None

(ii) Let U and V be finite dimensional vector spaces and T : U — V be a linear map one-one and onto, then :

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(a) dim U =dim V (b) dim U ≠ dim V

(c) U=V (d) U=d

(iii) Let W is subspace of vector space V. Then {f ∈ V/f(w)=0, ∀ w ∈ W} is called as :

(a) Hilatory of W (b) Annihilator of W

(c) Dual space of W (d) None

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(iv) The normalized vector (1, -2, 5) is :

(a) (1,-2,5) (b) (1/√30, -2/√30, 5/√30)

(c) (1/30, -2/30, 5/30) (d) None

(v) In IPS V(F) the relation ||u+ v ||² + ||u-v||²=2(||u||²+ || v ||²) is called as :

(a) Schwartz inequality (b) Triangle law

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(c) Parallelogram law (d) Bessels inequality

(vi) For two subspaces U and W of V(F), V=U ⊕ W & ..............

(a) U ∩ W= {0} (b) V=U+W

(c) U ∩ W={0} and V=U+W (d) None of these

(vii) Let T : M — N be an R-homomorphism. If B is a submodule then

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(a) T⁻¹(B) is submodule of N (b) T(B) is submodule of M

(c) T(B) is kernel of R-homomorphism (d) T(B) = T(M)

(viii) If T : U → V then the set {T(u) | u ∈ U} = ..

(a) Ker T (b) R(u)

(c) R(T) (d) None of these

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(ix) If || V|| = 1, then V is called :

(a) Normalised (b) Orthonormal

(c) Scalar inner product (d) Standard inner product

(x) If V is n-dimensional, then the dimension of V* is :

(a) Less than n (b) Greater than n

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(c) Equal n (d) Zero

UNIT—I

2. (a) Let U and W be two subspaces of a vector space V and Z = U + W. Then prove that Z = U ⊕ W iff z ∈ Z, z = u + w is unique representation for u ∈ U and w ∈ W.

(b) Extend the linearly independent set {(1, 1, 1, 1), (1, 2, 1, 2)} in V₄ to a basis for V₄.

3. (p) If U and W are finite dimensional subspaces of vector space V, then prove that : dim(U + W) = dim U + dim W - dim(U ∩ W).

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(q) Let R⁺ be the set of all positive real numbers. Define the operations of addition ⊕ and scalar multiplication ⊗ as follows :

u⊕v=u⋅v ∀ u,v ∈ R⁺

and α⊗u=uᵃ, ∀ u ∈ R⁺ and α ∈ R

Prove that R⁺ is a real vector space.

UNIT—II

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4. (a) If U, V is a vector space over a field F and T : U → V be a linear, then prove that : T(α₁u₁ + α₂u₂ + ... + αₙuₙ)=α₁T(u₁) + α₂T(u₂) + ... + αₙT(uₙ)

∀ uᵢ ∈ U, αᵢ ∈ F, 1≤i≤n and n∈N.

(b) Let T :V₄ → V₃ be a linear map defined by T(e₁)= (1, 1, 1), T(e₂) =(1, -1, 1); Te₃) = (1, 0, 0), T(e₄) = (1, 0, 1).

Verify Rank-nullity theorem.

5. (a) Find the matrix of the linear map T : V₂ → V₃ defined by T(x, y) = (-x + 2y, y, -3x + 3y) related to the bases B₁ = {(1, 2), (-2, 1)} and B₂ = {(-1, 0, 2), (1, 2, 3), (1, -1, 1)}.

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(b) Let U and V be vector spaces over the same field F. Then prove that function T : U → V is linear iff T(αu + βv) = αT(u) + βT(v), ∀ α, β ∈ F and u, v ∈ U

(c) If matrix of a linear map T with respect to bases B₁ and B₂ is :

-1 2 1

1 0 3

where B₁ = {(1, 2, 0), (0, -1, 0), (1, -1, 1)} and B₂ = {(1, 0), (2, 1)}, then find T(x, y, z).

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UNIT—III

6. (a) Let V be the space of all real valued continuous functions of real variable. Define T:V→V by

(Tf)(x)= ∫₀ˣ f(t)dt, ∀f∈V, x∈R

Show that T has no eigen value.

(b) Prove that if V be a finite dimensional vector space over F and v(≠0) ∈ V, then ∃ f ∈ V* such that f(v) ≠ 0.

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7. (p) If W₁ and W₂ are subspaces of a finite dimensional vector space V, show that A(W₁ + W₂) = A(W₁) ∩ A(W₂).

(q) If Kλ is eigenspace, then prove that Kλ is a subspace of vector space V.

(r) Define characteristic root and characteristic vector.

UNIT—IV

8. (a) In Fⁿ define for u=(α₁, α₂, ..., αₙ) and v=(β₁, β₂, ..., βₙ)

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(u, v) =α₁β₁ + α₂β₂ + ... + αₙβₙ.

Show that this defines an inner product.

(b) IF {x₁, x₂, ..., xₙ} be an orthogonal set, then prove that :

|| x₁ + x₂ + ... + xₙ ||²=||x₁||² + ||x₂||² + ... +||xₙ||²

(c) Prove that orthogonal complement i.e. W* is subspace of V.

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9. (p) If {w₁, w₂, .. wₙ} is an orthonormal set in V, then ∑ᵢ₌₁ⁿ |(wᵢ,v)|² ≤ || v||² for any v∈V.

(q) If V is a finite dimensional inner product space and w is a subspace of V, then prove that (W*)* = W.

(r) (i) Define inner product in vector space.

(ii) Define orthogonal set.

UNIT—V

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10. (a) Let A be a submodule of an R-module M and T is a mapping from M into M/A defined by Tₘ = A+ m, ∀ m∈ M Then prove that T is an R-homomorphism of M into M/A and ker T = A.

(b) Let T be a homomorphism of an R-module M to an R-module H. Prove that T is one-one iff ker T = {0}.

(c) Define :

(i) Submodule

(ii)) Unital R-module.

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11. (p) If A and B are submodules of M, then prove that (A+B)/B is isomorphic to A/(A∩B)

(q) Prove that arbitrary intersection of submodules of a module is a submodule.

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