Let A E R2x3, be R². Determine whether or not S = {x R³ | Ax=b, x₁ ≤ 0, x₂ ≤ 0} X2 sa convex set.

Q: ercise 6.67. Let E be a subset of R having at least two points. Pro- owing are equivalent:

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Q: 1 Determine whether or not the three given points le on a ne A (1,1,2), B(-1,-9, 3), C (3,2,1) 2.…

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Q: Draw a sketch of the following subsets of R2. {(x,y): x² +y² ≤ 1}.

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Q: Let A = the set of straight lines in the xy-plane. Let a L b mean “line a is parallel to line 6”.…

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Q: Let S = {(r.y) R²: (x-1)² + y² ≤4 and y> √3}. Question a. Find the interior, exterior and boundary…

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Q: 6. Given the universal set U ={x | x e 1, -10 sxS 10} A = {-2, -1, 0, 1, 2, 3 } B = {2, 3, 4, 5, 6,…

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Q: Let A = {1, 2, 3, 4, 5}. Define the following relations on A. R1 = {(1,1), (1,2), (2,1), (3,2),…

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Q: Let XC Rn convex. Prove that x is an extreme point of X if and only if the set Cis convex, where X =…

A: Let X⊂ℝn be convex . To prove: x is an extreme point of X if and only if the set C is convex, where…

Q: (E, Q4: state and prove L is a normed space ? e a ne. Vila a convex set.

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Q: Consider the set C = {x : Ax < b}, A e Rm×n and be R". Describe the set7 of feasible directions d at…

A: Given: The set C=x:Ax≤b, A∈ℝm×n and b∈ℝm   To describe: The set of feasible directions d at a point…

Q: Exercise. Give an example in R with the Eclidean space that is closed set but the union is not…

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Q: Indicate whether the following statement is true or false ? 1. If S = span{u1, U2, Uz}, then dim(S)…

A: If Span u1,u2,u3 then dim (S)=3 the  dim S=3 if S is spanned by u1,u2,u3 and  u1,u2,u3 are linearly…

Q: Consider the following linear map T: P3 → P3 defined as follows: T(ao + a1x+ azx² + a3x³) = ao +…

A: For T to be an isomorphism we must check that it is one one and onto both. And we can check it by…

Q: Let vị = V2 = V3 V4 and -[1} Confirm that p = V1 +v2 + ¿v3 + ¿V4 and vị – V2 + V3 – V4 = 0. Jse the…

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Q: Let U= {x:x E Z,0< x < 10}, A= { x:x EN,x² – 5x = -6}, B={ and C={c, d, e }: a=3 , b=0 , c=3, d=0 ,…

A: (a) Given : U=x:x∈ℤ, 0≤x&lt;10 which means all those integers which are greater than or equal to 0…

Q: get the following sets {(x, y) : x ∈[1, 2], y ∈[1, 2]} and {(x, y) : x = 2, y ∈[0, 1]}

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Q: Exercise 2.29. Pick several different subsets of R, and for each one, find its interior and boundary…

A: Suppose X,I is a topological space and A⊆X. Interior of A, Int(A) is defined as the union of all…

Q: If A⊂B, prove that if B is bounded, then A is also bounded. Is the converse true?(if not true give…

A: Let A⊂B, where B is bounded. Then there exist two real numbers k and K such that k ≤ b ≤ K for all b…

Q: 5. If r = (r, (x, y, z) and ro = (xo, Yo, 20), describe precisely the set of all points (r, y, z)…

A: We plug the given values in the given equation and simplify

Q: Q. Let A = {(x, y), y = x, x E Z,y e R}, Then A is a. connected and pathwise connected space. b.…

A: d. neither connected nor pathwise connected space.

Q: Solve number 1

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Q: Express the characteristics of the following sets of points, in terms of: closed or open, connected…

A: The set of points {z∈C: |arg (z)|&lt;(π/2)} is:   Closed: The set includes all complex numbers…

Q: 3. Use the definition of convex sets to answer the following: Show that if the sets S and T are…

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Q: Show that the set given in the image below is non convex. Please prove the result mathematically…

A: A set is said to be convex if for any two points in the set the line segment joining those points is…

Q: Let S = {[x y z] | 3x + 2y + z = 0} . Write the set S as the span of a set of two vectors in R 3

A: Let (x,y,z)∈Sthen 3x+2y+z=0                          z=-3x-2yTherefore,…

Q: Sketch the set of points satisfying: |z - 1 + 2i| = |z + 3 + i|

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Q: Find a topology in which the point x = 5 is an open set.

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Q: Let A € Rmxn and suppose that the rank of A is m where m is less than or equal to n and be Rm.…

A: Given: We are given a matrix A of size m×n with rank m where m≤n, a vector b of length m×1, and a…

Q: Prove the inequality |arctana−arctanb|≤|a−b|.

A: To prove the given inequality.

Q: Describe the set {TER²: doo (r,(0, 0)) = 1}.

A: To Describe: the set {x∈ℝ2: d∞(x,(0,0))=1}.

Q: A⊂B, prove that if B is bounded, then A is also bounded. Is the converse true?(

A: Let A⊂B, where B is bounded. Then there exist two real numbers k and K such that k ≤ b ≤ K for all b…

Q: A and B be two sets, each one of them having a minimum. Assume that A n B is not ty. Then An B has a…

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Q: Let fi = 1+x + x² + x³ , f2 = 1 + 2x + 3x? + 6x³, f3 = 3+2x + ? + 5x4, and g = 1+x + 2a? + tæ³. For…

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Q: Prove that |x|−|y|≤|x − y|. Hint: Apply the triangle inequalityto y and x − y.

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Q: The relation R = {(x, y): x + y s 2x} is defined on the set of integers. Which of th %3D following…

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Q: b. SSR ехту dA_ where R = {(x,y) | − x ≤ y ≤X, 0±× 6 2}

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