Let R be an integral domain and M an R-module. Define TorR(M): {me Mrm = 0, r0} (a) Prove TorR(M) is a submodule of M (b) Form the quotient module M/TorR(M) (recall the module action is r. (m+ TorR(M)) zero (i.e., it equals {0+ TorR(M)}) = rm + TorR(M)). Prove TorR (R/Tor (M)) is

Q: Please use typefont or legible handwriting with only the necessary steps and clear explanations.

A: Consider the provided question,

Q: 2 Find the area of the portion of the paraboloid x=9-y²-z² that lies above the ring 1 ≤y² + z² ≤ 9…

A: Given, The paraboloid y = 9-y2-z2  above the ring 1≤y2+z2≤9 in the yz plane.

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: assume that Pr[A]=0.25Pr[A]=0.25, Pr[B]=0.35Pr[B]=0.35, and Pr[A∩B]=0.1Pr[A∩B]=0.1. 1)Pr[A|B] =…

A: Given: P(A) =0.25 P(B)=0.35 P(A∩B) =0.1

Q: 1. Prove that if u and v are vectors in Rn, then |u·v| s ||u|||v|| where ju v| denotes the absolute…

A: As per guidelines we have to attempt only 1st question

Q: Seven thieves try to share a hoard of gold bars equally between themselves. Unfortunately, six bars…

A: Seven thieves try to share a hoard of gold bars equally between themselves.Unfortunately, six bars…

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: x · y = (x + y² – x² - y²)

A:

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: Find Im(A) for A = O Im(A) span = Im(A) = span Im(A) = span Im(A) = span 1 2 3 2 3 5 3 347 3 . Then,…

A:

Q: Fact: Z is a Euclidean Domain. What is a norm? Fact: R is a Euclidean Domain. What is a norm' Is Q a…

A:

Q: Let V be the set {(x, y) | x, y ≤ R} with addition operation defined by (x1, Y₁) ⇒ (X2, Y2) = (x1 +…

A: The addition operation is x1, y1⊕x2, y2=x1+x2, y1y2 and the scalar operation is α⊙x,y=x,0. First…

Q: Q8) The planes 3x+6z=1 and .2x+2y-z=3 are orthogonal F O

A:

Q: 2 2 6. State why = u,₁²v₁² + u₁₂²v₂² is NOT an inner product space for 2 u = (U₁₁U₂) and V = (V₁, V₂…

A:

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

A:

Q: 4. Show that the norm on l0 given by I| (11, .., In...) |lo= sup{|r;| : j= 1,2, ...} can not be…

A:

Q: 7. Consider the inner product defined on P₂ by (p, q) = p(-1)q(−1) + p(0)q(0) + p(1)q(1). f₁(x) = x…

A: It is given that,  p, q = p(-1)q(-1) + p(0)q(0) + p(1)q(1) be an inner product on P2. f1(x) = x-x2 ,…

Q: Claim that = (0.0.0) 2 a) show √ E OF √₂, √₂. b) Show OF J₁ () show J₁ s={√₁₁ √√₂, √√₂} S= √√₂₁ are…

A:

Q: (x1, y₁) + (x2, y₂) = (0,0) c (x, y) = (cx, cy) Determine if R² with these operations is a vector…

A:

Q: 4. (a) Find a bijection ƒ: (0, π) → R or prove that no such bijection exists. (b) Find a bijection…

A: (A). Bijection is f(x)= -cot(x)

Q: Show that 7(u, v)= (²-²,2ue) maps the triangle-((u, v):05≤SS 3) to the domain D bounded by z = 0, y…

A:

Q: Problem 2. Let U = {a+bx+ca²+dr³ € P3 (R): a-b+d=0}, 2(i) Find a finite subset S of P3 (R) such that…

A: