Let A and B be ideals. Prove that A + B = {a+ ß: a E A, ß E B} is an ideal.

Q: Prove that if a is perpendicular to b, then |a +b|^2 = |a|^2 + |b|^2 .

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Q: Find the characteristic of Z4 ⨁ 4Z.

A: Given: ℤ4⊕4ℤTo find: The characteristic of ℤ4⊕4ℤ.

Q: Let a=⟨3,−5,5⟩ and b=⟨−1,5,2⟩ Find the component of b onto a

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Q: complex numbers Z1= 4.5-j5.1 , Z2= 3.2- 5.2, Z3= 2.25-j5.6 , Z4= 7.5-j9.2 , then alculate…

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Q: 4. Find a monic polynomial of least degree in ℝ[x] that has 1-i and 2i as roots. write the powers of…

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Q: Que 405: Let S be a prime ideal of R and suppose that a product 1a2a3... an, of elements of R,…

A: Let S be a prime ideal of R and suppose that a product , of elements of R, belongs to S. Then prove…

Q: construct vw so that v= 2ab?

A: Here VW  and AB are line segments . To construct VW ,so that VW=2AB

Q: Suppose that x and y are real numbers such that y is 6 greater than x. What is the smallest possible…

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Q: Let A and В be ideals in principal ideal domain R , where A + R and В + R . Prove that the ideal AB…

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Q: How did they calculate what T does to the elements of B?

A: You want to know "How did they calculate what T does to the elements of B?" We have  B=1, x, x2, x3…

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: One of the following is an ideal of z1: (0,3} (0,2,4,6} None (9,3,6,9}

A: From given statement

Q: Ex: tet R = {o, a,b,c} a G O a b ( وه هه 2 2 G 6 ( 0 a C b a O a b a b 2/2 ( в с @ Show that (B,…

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Q: Given B1 and B2 as the bases of R?. B, = {[",]-C1} and B, = {G-C) %3D Find a. The coordinates of…

A: Ansa; Consider, 10=a11+b3-1⇒10=a+3ba-b⇒a-b=0⇒a+3b=0slove both equation and get;a=b=14 Co-ordinates…

Q: = 9. (15 points). Let S {0, 2, 4, 6, 8} C Z10. Is (S, +10, X10) a subring of Z₁0? Find the identity…

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Q: Prove that the point 2a +b-c, 5a -b+ 2c and 8a – 3b+5c are collinear.

A: Given, 3 vectors or points, A(2,1,-1) B(5,-1,2) C(8,-3,5)

Q: a) Calculate the 4th roots of (5-j3)

A: The objective is to calculate the 4th roots of (5-j 3).

Q: 3. Show that I = {0, 3, 6, 9} is an ideal of Z12. Show your work.

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Q: THE RADICAL OF AN IDEAL IS ALSO AN IDEAL.

A: We have to show that  RADICAL OF AN IDEAL IS AN IDEAL

Q: show by of R then IUI is not ideal example if In and Iz are icdeals an

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Q: Show that (AB)T and BTAT are equal.

A: Now,

Q: a b r s - Let R а, b, d 0 d r, s, tE Z, s and S even}.ir If S is an ideal of R, what can you say…

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Q: Explain why 3Z is not an ideal in Q

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Q: 1. Multiply. x' x x a. 5x'z x- 2xz2 - 4xy x-3x'yz g. 2a'b'c x 2abcd C. e. of

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

Q: 2. In the ring (4Z, +,.), the ideal (8) is (a) not prime (b) maximal (c) maximal and not prime (d)…

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Q: (c) Show that V3 e Q[ V6]

A: We have to proceed using contradiction method.

Q: Show that I+J is an ideal of L where I+J ={x+y:x∈I, y∈J}.

A: Given that and  are two ideals of a ring Ideal Test:A nonempty subset A of a ring is an ideal of…

Q: Consider the proof below of the fact that I + a = 1 +b iff a – bEI where R is a commutative right…

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Q: A (T+ 7) = A·I+ A ·G A (c.F) = c · (A · T) Verify these two properties by direct computation in the…

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Q: Consider the bases B₁ and B₂. B₁ = {[A], [4]},&₂ = ({[14]-[}}} 2 Find [x]82 if [X]8₁ = [₁]. 3

A: To find [X]B2, we can first find the matrix representation of the change of basis matrix from B1 to…

Q: Let a ER and let: 1 -2 2 1 = (1 ) A 1+ a 4+a -2-a a a You may assume without calculation that chÃ(x)…

A: Kindly refer to Step 2 for the solution.