8.F. Ifx= (1, I, --- ,)€R", define x, by e that z- l, is a norm on R'. Prove

Q: 8.0. If x-(x,, X, ...,%)«R', define [xL by Prove that xJkL is a norm on R'. N I|} dns - x]

A: We will solve the first question highlighted in yellow. If you want the second question to be…

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A: Use modulus property

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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

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A: Thje given functions are,u(x,y)=y(x2+y2) , v(x,y)=x(x2+y2)

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A: According to the given information it is required to prove that:

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A: Here we use basic real analysis

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A: I have find identity of the field and then inverse

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Q: 2/Define the topology If X-R² and T = {GK, KERZULAR} GK = {(XY) ER²|x+y >K3 Prove is topolgy on R²

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Q: defined BCX,x). a norm Dn

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Q: y Verify Euler's theorem for u = x' In

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Q: 1. Show that sinz is unbounded on C and lyl <|sinz] <ebl z=x+yi € C. for

A: Given:         We have to show that sinz is unbounded in ℂ and y≤sinz≤ey   for z=x+yi ∈ℂ Function…

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A: # given ui follow  uniform distribution (-theta,0) then to show that -min(ui) be the sufficient…

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A: Given that     ATAy=y  ∀y∈Rn. To prove ATA=In

Q: 2- Prove that: Jzt=27 dz %3D 2in. Whot is the princinle nart for f(z) = ze %3D

A: Since you have asked multiple questions in a single request we would be answering only the first…

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Q: Let 9,n be antegers wth 2n and ged (a,n)=1. Show tht of U(Z.) ={a*+nZ:tez} then o#1 mod 6) for all…

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Q: 4. Let xn → 0 and (yn) be bounded. Prove xnYn → 0.

A: Hello, thanks for your question but according to our guidelines I am answering just question number…

Q: Find the deriratire of y tanx usim the Difinition Rule

A: Given that:y=tanxTo find its derivative.

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Q: 32. Show that, in a normed space, x x implies -(x1 ++Xn) → x.

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Q: IRI- 2. <| sinz|< -e 2

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Q: - Pcosø prove that dw xy is Cont s Lony) - the fun ction (-1, 2) = x*y+z and x = PSinf cse y=pSi- -…

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Q: 2. Show using only the Cauchy-Schwarz inequality that if u,vE R", then || u- v ISl u || +| v | (1

A: Solution

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A: Suppose that xi and yi are two random samples of size n, drawn from a normal population.  Population…

Q: How do I show this proof?

A: Given that To show that,

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Q: 3,4 show that the integsal (Xy°ty%dx + 3) Independent of the path joining the peints (1,2)and 3,4).…

A: A function F is path independent if it is conservative. And if a function F is conservative then…

Q: The function u, takes on the following values : 1 1 1 1 1 u, = , u. + .....u, 10 1 + 3+1 + 32 +…

A: SOLUTION-

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A: Given that, f1(x)= x, f2(x) =cos2x and interval is [0, 90o] Wehave to find that the functions are…

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

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Q: 4-) Given a line l with direction vector and a plone jo with normal vecter Ñ show that ln=l or inP=Ø…

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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: Q3. Let z, y be real numbers such that ry 1 and z #y. Prove that

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Q: Show that Z/mnZ is isomphoric to (Z/mZ)×(Z/nZ) where gcd(m,n) = 1

A: Concept Used:( The First Isomorphism Theorem) If ϕ be the  ring homomorphism from G to G then…

Q: 10. Prove that the set H={[,ne subgroap of GL(2, R). For show that x HxcH but xHx+H. :9 GL(2,R),

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Q: If e-ispure imaginary W hat Restriction is placed on z?

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