Question 3. > Given the triple < Zx 3Z, +,- defined in terms of the Cartesian product Z x 3Z of two rings Z, and 3Z under operations (k, 3m) + (1,3n) = (k+1,3m +3n) and (k, 3m) - (1,3n) = (kl. 9mn) for k, l, m, n € Z. (a) Prove that < Z x 3Z, +, -> is a ring. Is it a commutative ring? Justify your answer. (b) Does the triple < Zx 3Z, +, > form an integral domain? Justify your answer.

Q: 2. Which of the following is a basis for the space of polynomials of degree < 2 over R? (Check all…

A:

Q: 2. Let z -1 + i √2 (i) Determine the modulus and the Argument of z. (ii) Locate z, 2², 2³, 24, 25,…

A:

Q: Explain why the operation (A, B)- tr(AB) defines an inner product on S, but not on Mn.

A: We know that M22 is the vector space of all 2×2 matrices with real entries and S2 is the space of…

Q: The curl of- yi+z+xR"

A: In this question, we find the curl of the vector function by using the determinant method.

Q: 4. Prove for A = {x: x = 12k, k = Z}, B = {x : x = 3k, k = Z}, and C = {x: x = 4k, k = Z} that A…

A:

Q: Show that for z, w € C: |z − w|² = |2|² + |w|² - 2 Re(wz). Sketch the set of complex numbers z EC…

A:

Q: 2. Let (X,(x) and (Y, (,)) be Hilbert spaces. For (x1,y1), (2, y2) = X x Y, put ((x1,y1), (2, 2))xxy…

A:

Q: 4) Define on the cartesian product of sets R:=R x R = {(^, z) | A, z E R } an addition and a…

A: We first show that R is a commutative ring. That is we show that R is commutative group under…

Q: 6. (a) Use the Cauchy-Schwarz Inequality to prove that, for any finite sequences of complex numbers…

A: All the problems are application of Cauchy Schwarz inequality

Q: Consider the set C4 = {i, -1, -i, 1}, the fourth roots of unity in the complex numbers. (a) Make a…

A: The given set is C4=i,-1,-i,1, the fourth root of unity in the complex numbers. We know that Cayley…

Q: b)Show that, 6(z, 3zy – y' and (z,y) = -+3ry are harmonie

A:

Q: 6 -3 1 Let (u, v) be the inner product on R generated by A = -1 and let u = (4, 3), v= (-5,3). Find…

A: Given that   &lt;u,v&gt; be the inner product on ℝ2 generated by A=61-1-3. And    u=(4,3), v=(-5,3).…

Q: . Show that ||(1, 2)||1₁ = |x1|+|x₂| is a norm on R², but that it does not satisfy the parallelogram…

A:

Q: write answer as fraction -(1/2)^3

A: We know , (a/b)n = an/bn

Q: True or False. The set {(0,0),(1,0),(2,0),(3,0),(4,0)} is a principal ideal of z5 x z3

A: Ideal:  A non - empty set S of ring R  is said to be an ideal of ring R if, (i)    a - b  ∈ S ;  ∀…

Q: Construct a idempotent of order 3, L3. Find L4 latin square of order 4, L4, and a commutative latin…

A: Given:An idempotent Latin square of order A commutative Latin square of order .Objective:Find the…

Q: Please solve 4,5,6,7 The Borel o-algebra on Ris generated by any of the following collections of…

A: As per the company policy, in a multipart question like this, we can solve only the first two…

Q: a.) Ker(p) and p(24) for p : Z → Z7 such that Ф(1) %3D 4 b.) Ker(p) and p(16) for p : Z → Z10 such…

A:

Q: 1. Find an r> 0 such that all zeroes of 24 + 6z²+z+3 are contained in D,(0).

A:

Q: (7.8) Lemma. Let K, L be simplicial complexes which intersect in a common subcomplex, then z(K U L)…

A:

Q: 20.* a. Let P(z) = 1 + 2z + 3z² +.. of P(2) are inside the unit disc. + nz"-1, By considering (1 –…

A: Proved by using Rouches theorem from complex analysis.

Q: TM) Suppese Ar and Bel a) and Be( e 2. if CCA) = CCB) then a is agqua l bo ? az 4/a=\/a=6 la=2?…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 1. Consider the nonzero complex numbers z, = 2i, zz = 8i and z; = 7i. (a) Evaluate (z,z,z,) (2) +2,…

A: Introduction: In most cases, the primary value Arg(z) of a complex number z=x+iy is equal…

Q: Find the area of JEN if the coodrinates of the vertices are J(0,3) E(6,0), and N(8,3)

A:

Q: 4. The field of quotients of (Z2, +2)-2) is (a) Q2 (b) Z2 (c) Z4 (d) Zg

A: Field of quotient

Q: 2. Complex Numbers: Show that for any two complex numbers, A₁ = a₁ + b₁j and A₂ = a₂ + b₂j, where…

A: The objective of this question is to prove the properties of the modulus and argument of the product…

Q: În a ring R, show that i) Oa = a0 = 0 ii) a(- b) = (- a)b=- (ab).

A:

Q: (8) In the ring (Z,, +6, •6 ), we get n{(a): a prime element in Z6} = ...... (a) $ (b) ((0), +6, 6 )…

A:

Q: . Let complex numbers a and 1/a lies on circles (x – xo)² + (y – yo)² = r² and (x- x)² + (y – yo)² =…

A:

Q: Find the centre and radius of the circle formed by all the points represented by z = x+ iy…

A:

Q: Show that (p(x), g(x)) = p(0)q(0) +p(1)q(1) + p(2)q(2) defines an inner product on P2.

A:

Q: X (5) Give an example of an abstract simplicial complex such that Geo (x) 2 is & 1-simplex in R²…

A: a) One example of an abstract simplicial complex X such that Geo(X) is a 1-simplex in R2 is the…

Q: 7. Let V={(x,y) x,y € R}. Suppose addition and scalar multiplication are defined using the following…

A:

Q: 2. (a) Design a bijection between ZU [1, +∞) and (0, +∞). Justify your answer.

A: To design a bijection between ZU1+x) and (0+x) according to the question, we can define a function f…

Q: 21. Show that the set of polynomials {C₁, C₂ (2x - 1), C3 (6x² − 6x + 1)} in P(R) is orthogonal with…

A:

Q: 6. Suppose that 1+2i is a complex zero of f(z)= z* – 2z³ +6z² – 2z +5, for all z eC then the other…

A: Given: The complex function is fz=z4-2z3+6z2-2z+5 for all z∈C. One of the root is (1+2i)

Q: Evaluate (2-)(2+항) Solution:

A: Complex multiplication,