1. Find the distance between the points (a) (4,-1,2,7) and (2,3,1,9); (-1,5,-3, 2, 4) and (2, 6, 2, 7, 6); (b) (c) (5,-2, 3, 3, 1, 0) and (0, 1, 5, 7, 2, 1). 2. For the given sets of vectors, find the dimension of the space spanne a basis for this space.

Q: If A+ iB is Hermitian (A and Bare real) show that [ AB -BA] is symmetric.

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Q: Theorem 2.1. For n ≥ 1 we have Σμ(α) d|n 1 -W-{ = 0 = if n = 1 if n > 1.

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A: note : As per the most general case we will assume the transition equation to be : pk = apk-1 +…

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Q: 2. For the vector field V = (7x³ + 2y2²)i + (4x²z - 7y²+3)j + (4y³ +2² -32²) k find the divergence…

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Q: (b) Find the multiplicative inverse of x + 1 + (x2 + 2x + 2) in Z3[x]/(x² + 2x + 2).

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Q: p = 39 is not prime. Find a number a that has no inverse z ( mod 39). This means that az = 1 (mod…

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Q: Show that every element of GF(pn) can be written in the form ap forsome unique a in GF( pn).

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Q: 8. The function f is given by the equation f(x) = (ar²+bx+c) e", where a, b and c are real…

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Q: 1. Let a, b, c be positive real numbers, and let p, q, r be [0,1/2]. Suppose a+b+c= p + q + r = 1.…

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Q: Show that the set K in the proof of Theorem 22.3 is a subfield

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Q: 3. What is the tangent plane to the surface 4x4 2xy + 7xyz -5 at the point P(1, 1, -1)? Find the…

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Q: If a and b are constructible numbers and a &gt;= b &gt; 0, give a geometricproof that a + b and a -…

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Q: Let p be a prime. Suppose that |Gal(E/F)| = p2. Draw all possiblesubfield lattices for fields…

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Q: . You have 10 identical cartons each holding 10 water bottles. All bottles weigh 10 oz. each, except…

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Q: Use Fourier-Legendre expansions to expand: f(x)=x² k = 3,4

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Q: find the general solution for Z_yy +Z=0

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Q: Prove or disprove that if D is a principal ideal domain, then D[x] isa principal ideal domain.

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Q: 1. modular arithmetic is the basis of many crypto-systems. Compute the result without a calculator.…

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Q: Evaluate the following integral : L² 1 (x³ + 2x) ex² dx

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Q: it needs matlab code  these functions in MATLAB code

A: Required Matlab code: syms x syms b lim((sin(bx))/x, b, inf)

Q: VII. Prove the following statements. 1. Let B be an orthonormal basis of a finite-dimensional inner…

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Q: Write all the expanded equations (12-31) and (12-32) for RP for a five-component system.

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Q: Straight lines passing through a fixed point (h, k). Parabolas with axis parallel to the x-axis.

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Q: B₁ = {V₁, V2, V3} and B₂ = {w₁, W2, W3} be ordered bases of R³. The change of basis matrix from Co…

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Q: If uHu = 1 show that I -2uuH is Hermitian and also unitary. The rank-one matrix uuH is the…

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Q: A quantum particle is trapped in a quantum system, giving it three discrete set of energy states.…

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