let S in Rn be convex set. show that u point in S is extreme point...

let S in Rn be convex set.

show that u point in S is extreme point of S if and only if u is not convex combination of other points of S.

Expert Solution

Colby Messinger answered 2 years ago

Let A ∈ Rn×n be symmetric and positive definite and let C ∈ Rn×m. Show:, rank(C^TAC)...

Let A ∈ Rn×n be symmetric and positive definite and let C ∈ Rn×m. Show:, rank(C^TAC) = rank(C),

Let x = (x1,...,xn) ∼ N(0,In) be a MVN random vector in Rn. (a) Let U...

Let x = (x1,...,xn) ∼ N(0,In) be a MVN random vector in Rn. (a) Let U ∈ Rn×n be an orthogonal matrix (UTU = UUT = In) and nd the distribution of UTx. Let y = (y1,...,yn) ∼ N(0,Σ) be a MVN random vector in Rn. Let Σ = UΛUT be the spectral decomposition of Σ. (b) Someone claims that the diagonal elements of Λ are nonnegative. Is that true? (c) Let z = UTy and nd the distribution of...

Problem 2.3 Suppose that we take the universal set U to be the integers. Let S...

Problem 2.3 Suppose that we take the universal set U to be the integers. Let S be the even integers, letT be the integers that can be obtained by tripling any one integer and adding one to it, and let V be the set of numbers that are whole multiples of both two and three. (i) Write S, T, and V using symbolic notation. (ii) ComputeS∩T, S∩V andT∩V and give symbolic representations that do not use the symbols S, T,...

Show that a set is convex if and only if its intersection with any line is...

Show that a set is convex if and only if its intersection with any line is convex. Show that a set is affine if and only if its intersection with any line is affine.

1: Let X be the set of all ordered triples of 0’s and 1’s. Show that...

1: Let X be the set of all ordered triples of 0’s and 1’s. Show that X consists of 8 elements and that a metric d on X can be defined by ∀x,y ∈ X: d(x,y) := Number of places where x and y have different entries. 2: Show that the non-negativity of a metric can be deduced from only Axioms (M2), (M3), and (M4). 3: Let (X,d) be a metric space. Show that another metric D on X can...

Let U and V be vector spaces, and let L(V,U) be the set of all linear...

Let U and V be vector spaces, and let L(V,U) be the set of all linear transformations from V to U. Let T_1 and T_2 be in L(V,U),v be in V, and x a real number. Define vector addition in L(V,U) by (T_1+T_2)(v)=T_1(v)+T_2(v) , and define scalar multiplication of linear maps as (xT)(v)=xT(v). Show that under these operations, L(V,U) is a vector space.

Let f : Rn → R be a differentiable function. Suppose that a point x∗ is...

Let f : Rn → R be a differentiable function. Suppose that a point x∗ is a local minimum of f along every line passes through x∗; that is, the function g(α) = f(x^∗ + αd) is minimized at α = 0 for all d ∈ R^n. (i) Show that ∇f(x∗) = 0. (ii) Show by example that x^∗ neen not be a local minimum of f. Hint: Consider the function of two variables f(y, z) = (z − py^2)(z...

. Let U be a non-empty set. For A and B subsets of U, define the...

. Let U be a non-empty set. For A and B subsets of U, define the relation A R B if an only if A is a proper subest of B. a. Is R reflexive? Prove or explain why not. b. Is R symmetric? Prove or explain why not c. Is R transitive? Prove or explain why not. d. Is R antisymmetric? Prove or explain why not. e. Is R an equivalence relation? Prove or explain why no

Let X be a set with infinite cardinality. Define Tx = {U ⊆ X : U...

Let X be a set with infinite cardinality. Define Tx = {U ⊆ X : U = Ø or X \ U is finite}. Prove that Tx is a topology on X. (Tx is called the Cofinite Topology or Finite Complement Topology.)

say R1, R2,...., Rn are commutative rings with unity. Show that U(R1 + R2 +.... +...

say R1, R2,...., Rn are commutative rings with unity. Show that U(R1 + R2 +.... + Rn) = U( R1) + U(R2)+ .... U(Rn). Where U - is the units of the ring.

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