Let (X, d) be a metric space and let A, B⊆X be such that A is connected, and A∩B ≠ ∅ and A∩ (X − B) ≠ ∅, prove that A∩∂ (B) ≠ ∅. Where ∂ (B) is the boundary of B.

Q: 4. Let V = R³ be a vector space with standard operations and vectors u, v € V. Define an inner…

A: Solve the following

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Q: 3. For each set below, circle "YES" it it is a vector space, or circle "NO" if it is not a vector…

A: First and last one has counter examples see the step 2 and second one is true

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Q: The taxicab metric p on the plane R² is defined by the rule: p((x1, y1), (x2, Y2)) = |x1 – x2| + |y1…

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Q: (X1, Y1, Z1) + (x2, Y2, z2) = (x1 + x2 + 3, Y1 + Y2 + 3, z1 + z2 + 3) с (x, у, z) %3D (сх, су, с2)…

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Q: Find the distance from y to the plane in R spanned by the set {u1, uz} 4 - 3 -3 y = -9 -4 Uz 2 4 1…

A: Solution   Given,       Find the distance from  γ to the plane  in R3       spanned  by the  set…

Q: The functions y(t) = et and y₂(t) =e are known to be solutions of y'' + Ay' + By = 0, where A and B…

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Q: 4) Let X and Y be normed Spaces.1f Tq:X →Y is a closed lincar operator and Tq:X →Y is a bounded…

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Q: 2) Let X be a vector space over IR and fa real valued fanctional on Such that (a) pexty) s pix)…

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Q: Show that set : A = {(x,y) = R² : sin(x) ≤ y ≤ 2e-2²] is closed on Euclidean plane R².

A: The given set is: A=x,y∈ℝ2: sinx≤y≤2e-x2. We need to shoe that the set A is closed in Euclidean…

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A: Let U=x,y: x2+y2=r2 : ∀ x, y∈R, r≥0 u1=x1,y1∈ℝ2:x12+y12=r12, r1≥0u2=x2,y2∈ℝ2:x22+y22=r22,…

Q: Let x = (13, −20, 15), v1 = (1, −2, 3), and v2 = (−1, 1, 1). Then the set {v1, v2} is orthogonal. It…

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Q: 8. Suppose I is a triangle formed by placing three points lircle, two of which lie on the circle's…

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Q: Is a subset of the plane {(x1, x2); x1 > 0, x1x2 ≥ 1} convex? Justify.

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Q: A.1 Which of the sets, equipped with the operations addition and scalar multiplication, are vector…

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Q: Let E R be a Euclidean space equiped with the dot product, i.e., for all %3D u = I2 and = in R, we…

A: As per our guidelines,  we are supposed to answer three sub-parts only. Kindly repost other part as…

Q: Give an example of (v1, v2, v3) and (w1,w2, w3) independent se- quences in the Rº such that…

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Q: Let V be a real inner product space, and let u, v, w E V. If (u, v) = 1 and (v, w) = 3, what is (3u…

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Q: Let V be the set {(x, y) | x, y ≤ R} with addition operation defined by (x1, Y₁) ⇒ (X2, Y2) = (x1 +…

A: The addition operation is x1, y1⊕x2, y2=x1+x2, y1y2 and the scalar operation is α⊙x,y=x,0. First…

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Q: Let V = R², defined the addition by: (U1 7 Uz) + (V1 7 vz)= ( U1 +V1 , Uz +Vz + 1) With standard…

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A: The equation of the plane is given by the formula ax+by+cz+d=0. The system of planes can be solved…

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A: Let u=u1,u2,⋯,un and v=v1,v2,⋯,vn be two vectors in ℝn. The sum of vectors can be defined as…

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Q: Prove that the norm induced by an inner product satisfies the parallelogram law. Using this…

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Q: - {z,y, 2)| 2 2= > 0} //| dpdød0 set Consider the 3D domain W = Calculate | a2 + y? + z² d(x, y, z)…

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