Let (X,Ox) and (Y,Oy) be affine varieties over K. Recall that we have constructed the affine variety X xY. Show that Oxxy(X×Y)= Ox(X)®Oy(Y).² 2.

Q: 6. The distance between a point x E R and a nonempty set A CR is defined by d(x, A) = inf {|x – y| :…

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Q: Show that the function a Y1 (x, y) = [21 2] в с Y2 is an inner product on R? if and only if b2 – ac…

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Q: If A = x a, – zy a,, evaluate § A · dī around the path L defined by 0 < x < 2,0 < y < 1 as shown in…

A: Stroke's theorem:         It is also known as Generalized stroke's theorem. It states that the…

Q: c) For x e ll n2, define || x ]| = || x || 1+ || x ||2: Show that (11 n2, ||· || is a Banach space.

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Q: b) Is there a linear map T: V → W, where dim V = 3, dim W = 4, rank T = 2, and dim ker T = 1?

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Q: 3;) Find the välie of k isó that. the furnictioin %3D .kx*. .if. x 2 2. is. continuous .cvery iw…

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Q: 4. Suppose that T : R* → R² is a linear map defined by T (x1, x2, x3, x4) = (2x1 x2 + x3, -2x1 – x2…

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Q: Let a < c < b and s(x) be a cubic spline with a single node c. Suppose s(x) = 0 on [a, c] Show that…

A: Cubic Interpolation Polynomial : Let  for  be the points satisfying the function . The cubic spline…

Q: 2. Find all volves of tand u x(t,4)=0 and simultomebusly such that =0. 3. 5x-11* C.

A: Get the equations for:  fx=0fy=0 15x2-11y=0-11x+3y2=0 or 15x2=11y3y2=11x

Q: Consider the closed curve C, which is th

A: Given, the closed curve C, which is the image of the path…

Q: Let (X, Ox) and (Y, Oy) be affine varieties over K. Recall that we have constructed the affine…

A: Given: Let (X, OX) and Y, OY be affine varieties over K. Now, we have constructed the affine variety…

Q: -1/2 8. Ž Pn (x) q^ generobr x,2) = (1-2xz + 2?) of legendre polynañials ,oblin the (2n+) xPa W=h+D…

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Q: 9. Show that the function (-, -) : IR² × R² → IR defined by setting = #1#2 + 2x12 + 2y1*2 + 8y1Y2 /2…

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Q: Prove that if Y:nz+P(x- mz) So thdz

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Q: a) Given ideals I, Jc R[r, y] show that the varieties satisfy V(In J) = V(I) UV(J)

A: Note:  According to bartleby we have to answer only first question please upload the question…

Q: Consider P2 with the inner product (p, q) = | p(x)q(x) dx . Let U = span{1,2x + 2} and g(x) = 3x? +…

A: Consider P2 (polynomials of degree atmost 2) with the inner product defined as follows, p, q=∫-11…

Q: Let p(x) =-2x* + c,x' + c,x + c,x + c, x ER be a polynomium whose graph insersects the points (-2,…

A: Given. We are given a fourth degree polynomial and also four points on the curve of the polynomial…

Q: Consider P2 with the inner product (p, q) = | | p(x)q(x) dx . Let U = span{1, 2x + 2} and g(x) = 3x2…

A: See the detailed solution below.

Q: find z such that 5.2% of the standard curve lies to the left of z.

A: Probability = 0.052 Finding the value of z corresponding to p=0.052= -1.62576 ( From Excel…

Q: (b) If X is a normed line- ball M= {xeX:x≤1}

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Q: - Show that the area of the triangle whose vertices are the [x1 yi det X2 Y2 A - points (x₁, y₁),…

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Q: The area bounded by the curve y = 7x and the lines z= 2 and =9 above the z-aris is O 539.00 square…

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Q: Prove that the curves o_1={|z| =2} and o_2= {|z-i|=5} are homotopic.

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Q: use the inner product = p(x)q(x)dx. to find the projection of p(x) q(x) = 5x + 2. = S'; x3 onto

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Q: Let f be defined and continuous on [a,b]. Then for each x E[a,b] there exists a polynomal p(x) such…

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Q: Let a and b be nonzero and f(x, y) = (ax + by)e-x Show that if a ± b, then there are five critical…

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Q: Prove that H(x, y) = (x, –y) provides a conjugacy between 1 X' = 1 X and Y' Y. 1 1 1 1

A: We are Given that :  H(x,y)=(x,-y) X'=11-11X , Y'=1-111Y The Matrix H is corresponding to the…

Q: Consider f : N → N, g : N → N and          h : N → R defined as f(x) = 2x,g (y) = 3y + 4 and h (z) =…

A: Given, Consider f : N → N, g : N → N and          h : N → R defined as f(x) = 2x, g (y) = 3y + 4 and…

Q: Show that if α and β are Dedekind cuts, then so is α + β = {r1 + r1 : r1 ∈ α and r2 ∈ β}.

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Q: 4. Let f : R" → R", x+ Ax + b be an affine function, where A E R"xn,beR". (a) Show that if X CR" is…

A: Let f: ℝn→ℝm be an affined map defined as x↦Ax+b, where A∈ℝm×n and b∈ℝm. Then it follows: fx=Ax+b…

Q: 6. The distance between a point x ER and a nonempty set A CR is defined by d(x, A) = inf {\x – y| :…

A: In this question, we have to prove that d(x,A)=0 iff x belong in closer(A)

Q: Suppose T E L(U,V) and S € L(V, W) are both invertible linear maps. Prove that ST E L(U, W) is…

A: First we prove that ST is a invertible linear map from U to W and then it will be shown that the…

Q: 3. Show that à = (6xy + z³)î + (3x² – z)ĵ + (3xz2 – y)k is irrotational. Find p such that A Vọ.

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Q: Determine the quadature rule of the form | f(r) dr = Aof(ro) + A1f(£1) + - *·+ Anf(Tn) a. when the…

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Q: Use Green's Theorem to evaluate F dr, where F(2, y) = (9xy, y" - 8) and C is the rectangle with…

A: That's easy. Have a great day!!!

Q: onsider two points (x0, y0) and (x1, y1). Prove that the first order Lan￾grange polynonial is…

A: There are two points (x0 , y0) and (x1 , y1) ,now prove that the linear Lagrange interpolating…

Q: Let C be a closed contour and zo be inside C. If P(z) is a polynomial of degree n, prove that P(z)…

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Q: Consider the following linear maps T : R² → Rª and S : Rª – M(2,R) defined as follows: 2x + 3y T(…

A: The solution is below.

Q: consider the map given by F(x)=1-x² discuss the charactristics of the orbits starting at x=0

A: As per the question we are given the following map : F(x) = 1 - x2 And we have to explain the…

Q: The curl of f(x,y,z)=xyzj is None of these. хуk-yzi О ху-yzk ООО О

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Q: Find the image of | z – 3i | = 3 under the mapping W =-.

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Q: Compute the value of f.dr If F= -yi+xj and c is a from (-1,1Y to CI,1) %3D SemICirele

A: Proceed as shown below

Q: IIC sUUset of R". tve R" and let k e R. Prove that S = {x e R":x.v = k} an affine subset of R". %3D

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Q: So F. dr - 0 for every closed curve Cin R, an open connected region. True False

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Q: Let F: IR3 →IR3 is a diffeomorphism and M is a surface in IR3 , prove that the image F(M) is also a…

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Q: 2 - Find the inter val on which f are Con Cave down and infleobion Points a -y = x2 5X+10 6- ソ= (¥+…

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Q: Show that if AT Ay = y for all y in Rn, then AT A = In.

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Q: Letf(z) 3z-2i. Find the maximum valne of /(=} on the disk z<.

A: Triangle inequality:z1-z2≤z1+z2 and z1+z2≤z1+z2

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.