Reference no: EM132238155

Assignment Questions -

Q1. Show that the set of rational numbers Q = {p/q, p, q ≠ 0 ∈ Z} is a field.

Q2. Show that the set of 3 × 3 matrices form a vector space over R.

Let S be the set of 3 × 3 matrices over R,

Q3. Let W be a subset of vector space V over K. ∀u, v ∈ W, α ∈ K, αu + v ∈ W , show that W is a subspace over K. Hence, show that the set of linear combinations W = {α₁v₁ + α₂v₂ + ··· α_nv_n, v_i ∈ V, α_i ∈ K, i = 1, n} is a subspace of V over K.

Q4. Show that the set of polynomials of degrees 0, 1, 2 and 3 form a vector space over Q.

Q5. Let V = {f: R → R} be the set of real number functions, show that V is a vector space over R. Define f, g ∈ V, (f + g)(x) = f(x) + g(x) and (cf)(x) = cf(x).

Q6. Show that vectors (x₁, y₁), (x₂, y₂) and (x₃, y₃) ∈ R² cannot be linearly independent.

Q7. Show that {u₁ = (a, b), u₂ = (a + c, b + c)} is a basis of V = R² over R.

a = {3, 4, 5, 6, 7}, b = {8, 9, 10, 11, 13, 17}, c = {14, 15, 16, 17, 19, 21, 23} mod(U, N)+1

Q8. Let {u₁, u₂, ··· u_n} be orthogonal vectors of vector space V.

(i) Show that {u₁, u₂, ··· u_n} are linearly independent vectors.

(ii) Show that the representation v = α₁u₁ + α₂u₂ + ··· α_nu_n is unique.

(iii) Given vector v, find its coordinates α_i (i = 1, n) with respect to vectors {u₁, u₂, ··· u_n}.

Q9. a and b are vectors in Rⁿ. Show that a · b = _k=1∑ⁿa_kb_k is a positive-definite scalar product.

Q10. f(t) and g(t) are continuous functions on [0, 1]. Show that (f, g) = ₀∫¹f(t)g(t)dt is a positive-definite scalar product.

Q11. Let V = Rⁿ, subspace S = {(a₁, a₂, ··· a_k, 0, ··· 0)} ⊂ V, find S^⊥ by the dot product.

Q12. Verify the Pythagoras theorem for vectors u = (a, b, c) and v = (b², c-ab, -b).

a = {3, 4, 5, 6, 7}, b = {8, 9, 10, 11, 13, 17}, c = {14, 15, 16, 17, 19, 21, 23} mod(U, N)+1

Q13. Prove the parallelogram law.

Q14. Find a_k for g(x) = x, a_k = 1/π _-π∫^π g(x)sin(kx)dx.

Q15. Show that (_i=1∑ⁿu_i²) (_j=1∑ⁿv_j²) ≥ (_k=1∑ⁿu_kv_k)², u_i, v_i ∈ R.

Q16. Verify triangular inequality with a 2D triangle ABC = {(0,0), (a, 0), (b, c)}.

Q17. Show that the size of a vector upon orthogonal projection is always smaller.

Q18. The scalar product of two functions f(x) and g(x) is given by (f, g) = 1/π _-π∫^πf(x)g(x)dx

Show that {sin(kx), k = 1,n} form an orthonormal basis.

Q19. Construct an orthonormal basis for vectors {(1, 2, a), (2, 3, b), (2, 5, c)}.

Q20. W = {(1, 0, 0, 0), (1, a, b, c)}, construct an orthonormal basis for W and W^⊥. Show that R⁴ is a direct sum of W and W^⊥.

Q21. Show that for linearly independent vectors {v₁, v₂, ··· , v_n}, an orthogonal basis can be constructed by the Gram-Schmidt orthogonalization process.

Q22. Show that the mapping d/dt : V → V is a linear operator, where V is the set of differentiable real number functions. What is the kernel of linear operator d/dt?

Q23. The eigenvalues of u and v are equal to λ, show that αu + βv ∈ E_λ.

Q24. If v₁, v₂ are eigenvectors of A with different eigenvalues λ₁ ≠ λ₂, show that v₁ + v₂ is not an eigenvector of A.

Q25. Show that the eigenvectors of a 3 × 3 matrix A with distinct eigenvalues are linearly independent.

Q26. What is the characteristic polynomial of matrix A = [a_ij]_n×n. Given A = find the characteristic polynomial of A.

a = {3, 4, 5, 6, 7}, b = {8, 9, 10, 11, 13, 17}, c = {14, 15, 16, 17, 19, 21, 23} mod(U, N)+1

Q27. Let {λ₁, λ₂, ··· λ_n} be the eigenvalues of a n × n matrix A.

Find α = λ₁ + λ₂ + ··· + λ_n and β = λ₁ λ₂ ··· λ_n

Q28. A = verify that P(5) = 0 and show that 5 is an eigenvalue of A.

Q29. A = find matrix B such that BB = B² = A.

Q30. Find the eigenvectors of symmetric matrix A =and verify that they are orthogonal.

Q31. Show that the eigenvalues of symmetrical 2 x 2 matrices are real.

Q32. Solve differential equations, dx/dt = bx - ay and dy/dt = cy - ax.

Q33. Show that q(x) = x^TAx is a quadratic form for n × n symmetric matrix A and n × 1 column vector x.

Q34. u, v, x ∈ V, compute (u, v) from the given quadratic form q(x).

Q35. Find the symmetric matrix A associated with the quadratic form q(x) = ax² + 2bxy + cy².

Q36. Find the range of q(x) = bx² - 2axy + cy² subject to the condition ||x|| = 2.

Q37. What is the range of Rayleigh quotient for matrix A = ?

Q38. Referring to Q18, verify the Pythagoras theorem with vectors u = sin(mx) and v = sin(nx) with m ≠ n.

Q39. What are similar matrices? Show that similar matrices A and B have same eigenvalues. Find a similar matrix to A = .

Q40. Let L be a linear operator on vector space V such that L(αu + βv) = αL(u) + βL(v), u, v ∈ V. Show that the kernel of L forms a subspace in V.

Q41. Find the least square fit to the following data. Estimate the value of y at x = (b + c)/2.

n	1	2	3	4	5
x	a	b	c	c+6	c+12
y	20	30	50	60	112

Note - Need Only First 10 Questions.

Attachment:- Assignment File.rar