a b Consider M₂ (R) {[%d) | a, b, c, d = R с X Y (a) Show that I = { [ + ² ] | x₁yER} is a subring of M₂ (R). Y X Z Z (b) Is J = ZER an ideal of I? Justify your answer. Z Z = }, a ring under matrix addition and matrix mutiplication.

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A: Since the question has multiple subparts, as per the company guidelines, we will solve the first…

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Q: 42. Rather than use the standard definitions of addition and scalar multiplication in R®, let these…

A: The given question can be solved as shown in step2.

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A: We have to prove that -

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Q: Show that the function defines an inner product for polynomials p(x) = a0 + a1x + . . . + anxn and…

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Q: ments is defined as N(a+bV2) = |a^2 ^2].Prove that for every a, bER (b 6= 0)it is sible to find q,…

A: Sol

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A:

Q: True or False: (p, q) = p(1)q(1)+p(2)q(2) +p(3)q(3) defines an inner product on P2(R).

A:

Q: Which one of the following is isomorphic to Z4 x Z6/((2,4))?

A:

Q: B = (f1, f2, f3) is a basis for R2[] where fi = 1+ 2x + a², f2 = 2+ 3r + 2x², f3 = 2 + x +3x². If…

A: The correct option is option (e) 1 + 2x²   The detailed solution is as follows below:

Q: ísomorp

A: Isomorphic to quotient ring:

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Q: QI/ 1- Write the multiplication table of the ring (Zs, +s,'s). 2- Is H = {0,2} subring of the ring…

A: To write the multiplication table for the ring (ℤ5,+5,, .5). Let   ℤ5=0,1,2,3,4

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A: We have given p=11-x+x2, q=7+9x2 Then, p=11-x+x2=112+-12+1212=121+1+112=123

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

Q: 3) Let Z, be ring integers mode 5 and f(x) € Zg(x) such that f(x)=x² + 2x + 1 then a) f(x) has only…

A: Since you have posted multiple questions, we will provide the solution only to the first question…

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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: Q3) Use the inner product = a,bo + a,b1 + azb2 to find , llp||, d (p, q), and then find the angle…

A: Given: Inner product: p, q=a0b0+a1b1+a2b2 and the polynomials  p(x)=1-x+x2 and q(x)=x-3x2. To find:…

Q: Use the inner product (p, q) = a b + a₁b₁ + a₂b₂ to find (p, q), ||pl|, |la||, and d(p, q) for the…

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