If U = span {x, y, z} in Rn, then show that U = span {x + tz, y, z} for all values t∈R.

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Q: Let X (2,3. 4) and de fined metrie on X as d(x, y) = |-y Find open ball B (2.2) = (0) O (2) 0 2.0…

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

A: Given X be a metric space with metric d. Now define a topology τ on X as follows: τ=B⊆X:…

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Q: If y is orthogonal to both and then y is orthogonal to every w in Span{u, v} V. O True False

A: We have to solve given problem:

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A: Given: Height of solid is defined by zx,y=16-x2-y2, R:    y=2x    and     y=4x-2 To find: Volume…

Q: part 4 of this problem

A: Introduction: Suppose A1, A2, …, An are n events, and A1ꞌ, A2ꞌ, …, Anꞌ are their respective…

Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

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Q: Given the bases B={u , u2} and B'={u'1, u'z} for R2, where , U2 and u'2 = The transition mat rix…

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Q: Let P, be an equidistant partition of [2, 4] on f(x) = x , then U(f, P,) =

A: To find the U(f, pn) of the function f(x) = x

Q: E be a Euclidean space of dimension (E(1, f)) = dim(Ker (f- id)), then s minimal.

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Q: OW Let w = f(x, y, z) and let x =psinocos0 and y = psinosin0 and z = pcoso. Find

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Q: Let (X,d) be a metric space. For x,y in X define e(x,y)=min{1,d(x,y)}. Prove that (X,e) is also a…

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Q: For every x, y E R, let p(x, y) = |x² – y²\. Is p a metric on R?

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: Let (X, Tp) be the particular point topological space with respect to p EX. Then for any subsetA S…

A: This is a question of Topology.

Q: Let u = and v = Show that is in Span (u, v} for all x and y. y -1 5

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Q: Let I = , zdV ,where D is bounded by the planes x = 0, z = 0, x+ y = 2 and the cylinder x2 + z2 = 1…

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Q: The set S = {T, (U) | U is open in X} U {T, (V ) | V is open in Y } is a subbasis for the product…

A: The given theorem is related with the topic product topology. Given that S = π1-1U | U is open in X…

Q: Let R have its usual topology and let X = {a,b, c}. Define f : R –→ X by а, х

A: Given, ℝ have its usual topology and let X=a, b, c. Define f: ℝ→X   by…

Q: Set (Q, M, µ) a space of o-finite measure. we define the aplication 4: L° (Q) → L'(Q)* given by =…

A: Let g,h∈L∞(Ω and α∈F Claim:- ϕ(αg+h)=αφ(g)+φ(h)

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: Let X = R, and d be the diserete metrie, and A= {1,2) Show that if A is compact or not

A: Given set A={1,2}

Q: be chown that y,= é" and can e are Jolutions to the diffierential ニ equatiune dy dy 4. w Cy, yz)= ?…

A: Given- It can be shown that y1 = e2x and y2 = xe2x are solutions to the differential equation d2ydx2…

Q: Let f E R[a, b] (which implies that f is bounded). 1. For any partition of a, b], show that U (2f,…

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Q: Let R? = RxR have the usual topology and let M = {(r, y) : 0 < x < 1,2 < y S 3}, then A. int(M) =…

A: Interior of a Set: Let be a topological space, and Then interior of a set A denoted byis the…

Q: Let X= [0, 0 ), and defined d as d(x,y)= e(xy) then verify all four condition of metric space

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Q: Let C(X,Y) have the compact-open topology. Show that if Y is Hausdorff, then C(X, Y) is Hausdorff

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Q: Convert the 2-column proof to a flowchart proof. Given: BD bisects ZADC and ZABC Prove: AABD = ACBD…

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Q: 3. Find the closure and interior of the set A = {(x, y): xy = 1} as a subset of R? (equipped with…

A: Closure and interior

Q: Compute the dimension of V,(Tỷ – Tỷ, TẬT, – T†T,) C P³.

A: Given that X=ϕ is a topological space. We have to compute the dimensions of the following.…

Q: Suppose d is Show that p: X?> [0,00) defined by a metric on X, p Cxy)= 3d (x,y) %3D 2+3d(x, u) is…

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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: If the region R is bounded between two circles x² + y? = 1, x + = 4, then we have: ev+ dr do dA = +…

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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: 4. Consider the space (IR,T), where T={R,Ø}u{AcR:[0,1)cA}. If A=(0,1), then the closure of A is:…

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Q: Define on R, the metric d defined by d(x, y) = |1+1리 1+ |y| Then (R, d) is not complete. True False

A: Given

Q: Use Green's Theorem to evaluate x dx + xy dy where C is the triangle with vertices (0,0), (1,4),…

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Q: Let E be the hemisphere described by Oszs V1-x? -y2 and F=Vz? + x2 + y² i+Vz? + x² +v² j+Vz? + x²…

A: We can solve this by gauss divergence theorem

Q: If X and Y are two independent RVs with f (x) = e*U(x) and f „(y) = e U(y) and Z = (X – Y)U (X – Y),…

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Q: „Consider the space (R,t). .where t={R,Ø}U{ACR:[0,1)CA} f A=[0,1], then the interior of A is (0,1) O…

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Q: ZVYW E ZTXZ, WX YZ, and ZZ ZW. Complete the proof that AVWY ATZX. W

A: Given: ∠VYW≅∠TXZ , ∠Z≅∠W Also, WX≅YZ   1 Adding XY to both sides of (1). We can do this using…

Q: For normal linear spaces X, Y, prove that BL (X, Y) is complete if Y is complete.

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Q: A subset W of R° is given by = {(x,y, z) | 2x = y = 2z}. (i) Write down the distinct elements of W.

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Q: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

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Q: GIVEN: LW 1TW, FX 1 XP, TF = PL, WL= XF. PROVE: AFML is isosceles. 1 2 F

A: To show triangle FML is isosceles. I.e. ,to prove MF=ML

Q: Let E be the upper half of unit ball E = {x2 + y2 + z2 ≤ 1, z ≥ 0}. Compute ∫∫∫Ez dV.

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Q: An Elliptical norm for R2 is defined by + 4u2, where x = (1,u). (i) Prove that l is a norm for R2…

A: This is a problem of Functional Analysis.

Q: Let X = {2,3, 4,..} and de fined metric on X as d (r, y) = |- yl %3D %3D Find орen ball B (2,2) =…

A: Follow the definition: B(2,2) = { x in X : d(x,2) &lt; 2 }

Q: Show that a triangle with vertices (x,Y;), (x2,y2), and (x;,y;) has area + x,y3 + X3Y, – X,Y3 – X,Y,…

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Q: Let X = {1,2,3, 4, ...} and de fined metric on X as d (r, y) |r-yl %3D Find open ballB (2,2) = o {2}…

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