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, n). (s, m) = (rs + rm + sn, nm) I 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 ((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: 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: Show that in the factor ring $Z[x]/\langle 2x + 1 \rangle$,the element $x + \langle 2x + 1 \rangle$…

A:

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

A: We have to find the .

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: Verify that the operation {x, y} = x1y1 − x1y2 − x2y1 + 3x2y2 where x = (x1, x2) and y = (y1, y2) is…

A: Given : The operation on ℝ2 is defined as  x, y = x1y1 - x1y2 - x2y1 + 3x2y2 ,  where x = (x1 , x2)…

Q: Q1) Let D = M₂ (Z2)be a ring find the order of D and all the idempotent elements.

A:

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

A:

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

A:

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: Explain why is not an inner product for u =(u1, u2) and v =(v1, v2) in R2 when = u1u2 + v1v2 using…

A:

Q: (†) \u,v≤U ([u‡ 0 and v‡0] → u • v‡ 0); (⋆) Vx,y,z€U ([z ‡ 0 and zx = zy] →x=y).

A:

Q: 2. Determine if the function (u, v) = µ₁ V₂ + U2V1, where ủ = (u₁, U₂) and ở = (V₁, V₂) defines an…

A:

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

A: We have to find