4) Define on the cartesian product of sets R:=R x R = {(^, z) | A, z E R } an addition and a multiplication as follows: (A, z) + (µ, w) := (A+µ,z +w) and (λ, 2) . (μ, w) :(μ, λυ + μ). Show that: (i) R with these two operations is a commutative ring; (ii) R, {(0,0)}, and I:= { (0, 2) | z € R} (0, z) are the only ideals in R; and (iii) I = Nil(R).

Q: Consider V = {(2,y) : x, y ≤ R}. Let and ⋆ be operations on this set defined in the following way:…

A:

Q: (Q. +,.) is a subfield of (F = {a + bV5: a, b e Q}, +,.); (F = {a + bv5: a, b e Q}, +,.) is a…

A: A non-empty subset F' of a field F is a subfield of F if F' is closed with respect to the…

Q: (a) Let M = R. Give the radius r and the center c of B(-2,5) n B(6, 7). ne

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: If A and B are orthogonal, is A+B also orthogonal?  Can you demonstrate Algebraically using A + B =…

A: Solution: Given-   A and B are Orthogonal matricesTo show:  (A+B)  is OrthogonalSince, A and B are…

Q: for the discrimanat here dont we do (Uxy)^2-(Uxx*Uyy)

A: Please see the below picture for detailed solution.

Q: (2) Write v as a lincar combination of u₁, U₂, U3 if possible, where U₁ = (2,3,5), u₂ = (1, 2, 4),…

A: Given that,  V=(10, 1,4)  u1=(2, 3,5), u2=(1, 2,4) , u3=(-2, 2,3)  To find : a, b, c such that…

Q: 2) Let R real numbers, define = x,Y1 +4x2y2 + 6x3Y3 for x = (x1,x2, X3) and y (y1,y2, Ya). Is inner…

A: Let V be a vector space over ℝ.An inner product <·,·> is a function V×V→ℝ with the following…

Q: 25) Which notation describes the translation of the sailboat B(70, 20) to the lighthouse B'(10, 80)…

A: Given: B70,20, B'10,80 For reaching from point B to point B', we have to use operation x,y⇒x-60,…

Q: Given that x, y, and z . Prove that if (x^2)|y and (y^3)|z, then (x^6)|z

A: Solution: The objective is to prove that the given statement

Q: xʻy Let f (x, y) = if (x, y) # (0,0) and f (0,0) = 0. (x4+6y8) r4.

A:

Q: Let V3 = {(a, b)|a, b E R} with the following operations: (a₁, a₂) + (b₁,b₂) = (a₁ + b₁, a₂ + b₂)…

A:

Q: FOR ANY TWOCOMPLEX NUMBERSZ, ANDZ,,PROVE THAT |Z|+|Z,|=,(Z, +Z,)+ /Z,Z» +÷(Z, +Z»)- Z,Z, .

A: for two complex number Z1 and Z2 prove that Z1+Z2=12Z1+Z2+Z1Z2+12Z1+Z2-Z1Z2

Q: 5 - 25 + Z² = -1 2x + 34 = 2 -y-27 = 0 (Mabrices), find of 2 Rom Rescaling

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Q: Show that  ℝ2 = span([1 1], [1 -1])

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Q: G) In 2D transformations, is the following relation TRUE or FALSE? [2] R(9, )R(9, )= R(9, )R(9, ) =…

A: The given relation is true regarding the theorem of transformation in case of 2D

Q: 30²) For A = {• t, 1} & B = {₁, 0, 83 & universum U= {0, oto, 1, 0, 7, 3 Find: A', B', AUB, AnB, AB,…

A: (i) Consider the universal set is U=∘,+,1,⋄,7,Ω, A=∘,+,1 and B=1,⋄,Ω. (ii) Given U=ℝ×ℝ=ℝ2, A=x,y|x≥1…

Q: how that ℝ3 = span( 1 1 2 1 2 1 0 3 -1

A: We want to show that  any vector can be written as linear combination of the three  given vectors…

Q: Please compute x and y such that 117x + 17y = gcd(117,17), 52x + 56y = gcd(52,56), and 123x + 126y =…

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Q: Compute (2sin z -) ) da Show work but don't simplify.

A:

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: Let u = (-1, 1, 0) and 7 = Compute the following: ú.v= v. ú= = (-7, 6,-5). =

A: “Since you have posted a question with multiple sub-parts, we will solve first three sub parts for…

Q: Find a number z (except 0) such that Az³ + Bz²+ Cz = D 2 1 3 0 -2 -4 00 ^-[ ] ³-R2 :]-[+]-[1] B= 1 2…

A: Solution

Q: Find the cross product ax b: a=(2, – 1, 2), b=<0, 4, 5)

A: Given that: a=&lt;2, -1, 2&gt;b=&lt;0, 4, 5&gt;