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 + n) (r, n). (s,m) = (rs + rm + sn, nm) show that (R × Z,+,.) is a ring:

Q: Let (u, v) be the inner product on R² generated by A = [ 6 -1 and let u = (3, −5), v = (7,4). Find…

A: We have to find <u, v>

Q: Let V=R². For (₁, ₂), (V₁, V₂) € V and a ER define vector addition by (41, 14₂) H (V₁, V₂) := (U₁ +…

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

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Q: Let V = R². For (₁, U₂), (V₁, V₂) € V and a ER define vector addition by (4₁, U₂) (V₁, V₂) := (U₁ +…

A: Let For  define vector addition by  and scalar multiplication by .

Q: Find (u, v), ||u||, ||v||, and d(u, v) for the given inner product defined on Rn. U= (0, -3), v =…

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Q: If u xv=uxw and u ‡ ♂, then u = w.

A: The given statement is: If u×v=u×w and u≠0→, then u=w

Q: Let ((u,,u2,U3),(V1,V2,V3))= 2u, V1 + 3u2V2+ U3V3 be an inner product on R. Determine ||(1, – 1, -…

A: Explanation of the answer is as follows

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: 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: 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: Find (u, v), ||u||, ||v||, and d(u, v) for the given inner product defined on R". u = (5, 0), v =…

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Q: Find (u, v), ||u|l, ||v||, and d(u, v) for the given inner product defined on R". u = (3, 0, 1), v =…

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Q: Show the span {(1,0, 1), (-1, 2, 3), (0, 1, – 1)} is all of R'.

A: Let x=1, 0, 1, y=-1, 2, 3 and z=0, 1, -1. We have to show that spanx, y, z is actually the vector…

Q: Find projvu and projuv. Use the Euclidean inner product. u = (7, -5, 1), v = (1, -1, 0) (a) projvu…

A: The given vectors and .We have to find the values of the projection,a) b) .

Q: Let (u, v) be the Euclidean inner product on R², and let u = v = (3,5). Compute the following. NOTE:…

A: Given that the vectors  and .Using the Euclidean inner product on  to find the  and .

Q: Let { = fo + ū and 7 = no + ở so that quaternion multiplication gives §n = £ono + £oð + noũ + ữỡ.…

A: are ξ,η are quaternion numbers So, Let  ξ=ξ0+iξ1+ξ0+Iε2+kξ3 u→=iξ1+ξ0+Iε2+kξ3 and η=η0+iη1+Iη2+kη3…

Q: Which of the following are bases of R3? {(1, 0, 0), (1, 1, 1), (2, 1, 3), (1, 1, 1)}; {(1, 0, 0),…

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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: The group of all rotations in space that preserve a given cube has order: (u) 6. (b) 12. (c) 8. (d)…

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Q: Let V = R?. For (u1, u2), (v1, v2) e V and a e R define vector addition by (u1, u2) # (v1, v2) :=…

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Q: True or False: (p, q) = pq defines an inner product on P(R).

A:

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

A: Given problem is :

Q: Use the Chain Rule to compute

A: The partial derivative is ∂∂ulnu2+uv.   The partial derivative of a function with respect to a…

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 →a=⟨−4,1,−2⟩a→=⟨-4,1,-2⟩ and →b=⟨−2,−3,4⟩b→=⟨-2,-3,4⟩. Find the projection of →bb→ onto →aa→.…

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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: Prove Theorem .Let Q be an orthogonalmatrix. detQ=+-1

A: Given that, “Q” be an orthogonal matrix, So, QQT=I=QTQ…

Q: Suppose that c is the line segment from (a, b) to (c,d). (a) Show that [rd |x dy – y dx = ad – bc…

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Q: What is the Cartesian Product of Z × X × Y, where X = {3,–1}, Y = {m, l, 0} and Z {w,b}.

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Q: Which of the following sets form bases: ) (1,1),(-1,1) for R² (1,2,1),(-1,2,1),(0,4,2) for R³…

A: Here we have to check whether the given set of vectors form a basis or not.