Let R be a ring and consider RxZ={(r,n)|r e R, n e Z}. Define (r,n) + (s,m) = (rts, n+m) (r,n)(s,m) = (rs+mr+ns, nm) Show that R x Z is a ring.

Q: Let R be a ring and consider R x Z = { (r,n) | r e R, ne Z} Define (r,n) + (s,m) =(rts, n+ m) (r,…

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Q: 3. Compute the inner product g (v,w) of v = + x. 2 əx and w = x.-+ 2y. Əx ay' Where the line element…

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Q: For the following fields, calculate div F. 1. F (x,y,z) = 〈2xz, -3xy, -z2〉 2. F (x,y,z) = 〈3x+8,…

A: Step 1:Step 2:

Q: {6 ):a € Z,6b,c e Q}. Together with matrix addition and multiplica- Let R= tion, show that Ris a…

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Q: 3. Let z = (21, 22), w = (w₁, W₂) E C². Show that (z, w) = z₁w₁ + Z₂W₂ is an inner product on C²

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Q: Let (R, +,-) be a ring and a, b E R. Prove the following. *)If ab = -ba, then (a + b)² = a² + b².…

A: Ring and integral domain

Q: Let (R, +,.) be any ring. Define R × Z = {(r,n):r E R ,n E Z} and %3D (r, n) + (s, m) = (r + s,m +…

A: In the Structure T=ℝ×ℤ additional multiplication is defined as…

Q: Let (R, +,.) be any ring. Define R x Z = {(r,n):r ER ,n E Z} and (r,n) + (s, m) = (r + s,m + n) (r,…

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Q: Find (u, v), ||u|l, ||v||, and d(u, v) for the given inner product defined on R". u = (1, 1, 1), v =…

A: Our guidelines we are supposed to answer only three subpart. Kindly repost other subpart as the next…

Q: One of the following is not integral domain a) (Z, +7,7) b) (R, +,) c) (C,+,) d) (P(X),A,0) a) O d)…

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

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   <u,v> be the inner product on ℝ2 generated by A=61-1-3. And    u=(4,3), v=(-5,3).…

Q: 9. Show that the function (-, -) : IR² × R² → IR defined by setting = #1#2 + 2x12 + 2y1*2 + 8y1Y2 /2…

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Q: Use the definition of an inner product to determine if "⟨a, b⟩ = a′(0)b′(0) for a,b ∈ M(−1, 1) " is…

A: This is a problem of Functional Analysis.

Q: Let R is ring of real numbers and *, (2) O defind on R^2 as follows, (a, b), (c, d)ER^2 then (a, b)*…

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Q: Let Consider the complex inner product space C³ with the usual inner product (u, v) = ₁V₁ + U₂ V₂ +…

A: Inner products space

Q: Let R is ring of real numbers and *, (2) O defind on R^2 as follows, V (a, b), (c, d)ER^2 then (a,…

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Q: 1. Let R be the ring of all functions f: R → R, with operations of pointwise addition and…

A: Ideal:  Let  is a ring. A nonempty subset  of a ring  is call an ideal of , if following are true.…

Q: (a) (b) Let V = F" and let A EMnxn (F), F is complex field. Prove that (x, y) = (4¹x, y) for all x,…

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Q: Consider the inner product on R2 defined by ((x, x2), (y, y2)) = x11 + 4x2y2. Which of the following…

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Q: State why (u, v) is not an inner product for u = (ui, u:) and v = (v, V2) in R? %3D (u, v) = (u,v;)²…

A: We are given that u,v=u1v12+u2v22 for u=u1,u2 and v=v1,v2 in R2. We know that an inner product must…

Q: Show that the following functions define an inner product on (R)2 where U= (x,y) and V= (x2,y2) a)…

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Q: Is = xjx + yjy+zźz3 an inner product on R'?

A: No it is not an inner product.  Inner product looks like a dot product.

Q: Let ((uz,u2,U3),(V1,V2,V))= 2U1V1 + 3u2V2+ U3V3 be an inner product on R, Determine d(1, – 1, –…

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Q: Let S = {a+b√2| a, b ≤ Q}. Is (S, +, ×) a ring? Prove your result.

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Q: Prove that if (I,+,) is an ideal of the ring (R,+, ), then rad I In rad R. %3D

A: The term radical is used when we think about ideals and when we talk about ideals definitely…

Q: Let ((u1,u2,U3),(V1.V2,V3))=2u1 V1 + 3U2V2+ U3V3 be an inner product on R3. Determine ((1, – 1, –…

A: Given problem is :

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

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Q: 3. A finite-dimensional Hilbert space is a finite-dimensional complex vector space H together with…

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Q: 9. Let D be an integral domain such that (x + 1)² = x² + 1 in D[x]. Char D = а) 0 b) 1 c) 2 d) 3 O…

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Q: For (a, B) E R?, define a function (-, -) : R² × R? → R by setting arr' + xy + Yx' + Byy' . 1. Find…

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Q: Let (u, v) be the inner product on R² generated by A = 1=[ A and let u = (3,-4), v = (-4,5). Find…

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

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Q: Let V be the set V = {(x, 1, z)| x, z E R}. Define the addition and scalar multiplication on V by…

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