Let v e R" and letk e R. Prove that S = {x € R" :x•v = k} is an affine subset of R". %3D

Q: 69. Let F(x, y) = (x,0). Prove that if C is any path from (a, b) to (c, d), then F. dr (o2

A: Given: F(x, y) = <x, 0>. C is any path from (a, b) to (c, d).

Q: Let G = Z31, S = {1,5, 6, 25, 26, 30}. %3D %3D Prove that Cay(G, S') is normal.

A: It is given that

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A: Solution:-

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

Q: If there exists open ball B (a, r) such that B (a, r) NG= Ø then a is not limit point of G True O…

A: TRUE

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: is absolutely and uniformly conyergent for all vlues of x an t in the circle 121 <B-¹

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Q: 6. If Z is the ocentroid of AWXY, WR = 87, SY = 39, and YT 48, find each missing measure. W a) WS =…

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Q: Let A = psino a,+p a,. Evaluate A dl given that L is the path shown in the figure. 1. Verify your…

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Q: If there exists open ball B (a, r) such that B (a,r) nG = Ø then a is not limit point of G True…

A: TRUE

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: 25 If A E M, is symmetric, prove that AT is symmetric.

A: Symmetric matrix

Q: In which of the following cases is the mapping x: R² → R³a patch? (а) x(и, v) — (и, uv, v). (c) x(и,…

A: A mapping defined from x : R3 to R2 is defined a patch if x maps U diffeomorphic ally onto x (U) As…

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

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Q: 1 Show that IS the separable me tre space (R, d), where el(x, y) = be-y/

A: Here we use basic real analysis

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: Consider the region D in t

A: Given, the curves                 y=2x, y=4x, y2=x, and y2=3x To find ∬Dx2x4dA

Q: Q/ show thet f(2) = 2+ Conformal maps 121= 1 ? on

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Q: Let D be a Euclidean domain with measure d. Prove that u is a unitin D if and only if d(u) = d(1).

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Q: 3. Let r>0 and define the closed ball by B,(1) = {y €X : d(r,y) Sr} Show that B,(r) is elosed set…

A: Given that X be a metric space with metric d. The given topology on X is defined as:…

Q: Find the image of the infinite strip <ys under the trans-

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Q: Let, T:R' → R';T(x,y,z) = (x + y+ z,2y+z,2y+3z) 2. Find the dimension of KerT and ImT

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Q: 3. Use Green's Theorem to evaluate G³ï+xj)•dr around the perimeter of the rectangle 0<x<2,0<y<3.…

A: Given ∫C(y2i+xj).dr We know that dr = dxi+dyj Thus ∫C(y2i+xj).dr = ∫C(y2i+xj).(dxi+dyj)  =…

Q: Let T : V --&gt; W be linear, b ϵ W, and K = {x ϵ V: T (x) = b} be nonempty. Prove that if s ϵ K,…

A: Given: Let T : V --&gt; W be linear, b ϵ W, and K = {x ϵ V: T (x) = b} be nonempty The objective is…

Q: Given any Hermitian space E with › E, if (f(x), x) = 0 for all x € E, th I

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Q: c If in is an involution of X, show that (r, y)in = (yi", rin) is an involution of X12). %3D

A: Given X is a lattice. Given in  is an involution of X To show: x,yin=yin,xin is an involution of X2…

Q: In a metric space (X,p), if lim un = u, lim v, = v, show that lim d(un, Vn) = d(u, v). %3! n-00

A: Convergent Sequence: A sequence xn in a metric space X, d is said to converge to a point x∈X if for…

Q: et S be a nonempty subset of R and let k E R. Define kS {ks : s E S}. Prove if k > 0, en sup(kS) = k…

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Q: Let F(x, y) = (x, 0). Prove that if C is any path from (a, b) to (c, d), then F dr = 3 – a²)

A: Consider the provided information, Let Fx,y=x,0. Prove that if C is any path from a,b to c,d then…

Q: Given the bases B= {u,,u2} and B'= {u'1, u'2} for R, where U2 and The transition mat rix Pg-g from B…

A: We first find the coordinate vectors of u_1 and u_2 relative to basis B' ,then we will form the…

Q: PO しl Let 4Let R be the reaion enclosed by oe en closed by Jhe graphs [sreph R ont omt y=8; x= o ind…

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

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 P = {(z, y, 2) E R° | az + by + cz = 0} where (a, b, c) # (0,0,0). (a) Prove that (a, b, c) ¢ P.…

A: Given that P=x,y,z∈ℝ3 : ax+by+cz=0 where a,b,c≠0,0,0 (a) The objective is to prove that a,b,c∉P Let,…

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: Given: mAB = x Prove: mz1 = 180° - x A D Exercises 25, 26

A: Given:           m AB⏜ = x   To prove:              m&lt;1 = 1800 - x

Q: Consider R? and R3 with the Euclidean (c) norms. Compute || T || for T : R2 →R3 defined by T (x1,…

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Q: 3. DETERMINE THE UOLUME OF THE REGION BOUN DED ABOUE BY THE PARABOLOID Z=2-x²-y BY THE CON E

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Q: u* (B) = inf{l&|: is a covering of B by closed boxes} for all BcR" %3D صواب İhi O

A: I will use the definition of measure.

Q: Consider IR and define Tę and E where I is the tepolagy cofinite topology and Ic is countable then…

A: Consider τp is co-finite topology and τc is countable topology. Assume, τp to be, a,b|a&lt;b And τc…

Q: Qsit Prove that the Centre of a gp îs noymal. of

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Q: Let A = psin ø a,+p a,. Evaluate PA dl given that L is the path shown in the figure. Verify your…

A: It is given that, A=ρsinϕ aρ+ρ3 aϕ   According to stokes, theorem (also called Curl theorem),…

Q: let Z be aranelom vanimble with Pelf (Z~Nco) Show that , the psf of Z² (Y=Z) is. chi-square

A: # Given : z be the standard normal variate  let y=z^2  then to show that y follow chisquare…

Q: O let T:R?→R² be a lineas tanafor- mation Such that T([:])- [ ]. T %3D Find matnie A Such that T(x)=…

A: Given that T:ℝ2→ℝ2 such thatT11=41  and T01=32 We have to find the matrix A such that Tx=Ax   ,…

Q: Let X = R, and let T consist of 0, R, and all intervals of the form (a, b) where a, b e R and a < b.…

A: A topological space is an ordered pair {\displaystyle (X,\tau ),} where {\displaystyle X} is a set…

Q: Let A EMn (R) be given. If 11x11 = 11yll = 1 and x •y = 1, show that x=y

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Q: A mapping f:C→ C is called bounded, if sup |f(2)| < o0. zEC Show that sin, cos : C →C are unbounded…

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Q: Show that every covering of the Möbius strip is homeomorphic to either R², S¹ x R, or the Möbius…

A: We have to prove that every covering of the Mobius strip is homeomorphic to either R2or the strip…

Q: 3.1 Show that upper bound of fe dz, where C is the circle |z + 4 – il = 7. z+5i-1

A: The given problem is to find the upper bound of the given complex integral over the region of circle