Show that a Hilbert space is closed. Also, for any square-integrable random variable X, show that L2(N, σ(X), P) is a closed subspace of L₂(N, F, P).

Q: Find a basis for the subspace of R3 spanned by S = {(2,3, -1), (1,3, -9), (0,1,5)}

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Q: The trace of a squaren x n matrix A = (aj) is the sum a₁1 + a22+...+ann of the entries on its main…

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Q: What is the dimension of the subspace or the space of 5 x 5 matrices (ai) sucn that a11 +...+ az5 =…

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Q: Let X be a Gaussian random vector with distribution [5 4 ~~ (B]. []). 5 X~N Let Ỹ be a Gaussian…

A: Given:Mean vector of X is:

Q: M2x2 (R) is a vector space over R. Make use of this result to show that the subset S defined by a S…

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Q: The set S is {F}} 2 -2 5 neither an orthogonal set nor an orthonormal set. an orthogonal set. a…

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Q: (b) For subspaces X and Y, the sum X + Y is a direct sum (denoted X + Y) if every VEX + Y can be…

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Q: Suppose W,, W2,. (a) span (W, W2 (b) If S; spans W; for i = 1,...,r, then S, US, U...US, spans W, +…

A: (a) W1+W2W1+W2 would be W2W2. Below is the justification.  x+y∈W1+W2,wx+y∈W1+W2where x∈W1x∈W1 and…

Q: Consider the subspaces U = {(x,x) | x ≤ R} and W = {(0, y) | y ≤ R} of R². Show that U + W is a…

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Q: Given U_1, ..., U_m, a collection of subspaces of some vector space V, recall that U_1 + ... + U_m…

A: Here, in the question, there is a collection of subspaces given. We have to give an example of the…

Q: Prove that if a 1-dimensional subspace W of Rn contains a nonzero vector with all nonnegative…

A: Let v be a nonzero vector with all non-negative entries in W and let d be the sum of all of its…

Q: Let U and V be the subset of R' defined as U = { (a,b.c,d) : 3b-2d-a = 0 } and / = { (a,b,c,d) : a=c…

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Q: Either prove or disprove that the set V is a subspace V ⊂ R^n defined by  V={x: Ax=λx}where A is…

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Q: Let M3,3 denote the vector space of all 3 × 3 matrices. Let L3,3 denote the set of all lower…

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Q: Find a basis for the subspace of R4 spanned by S. S = {(-11, 6, 4, 8), (-6, 3, 2, 4), (-2, 1, 1, 2),…

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Q: H is a Hilbert space 1. Let M be a closed subspace of H. Show that (M)+ = M. %3D

A: Given that M is a closed subspace of a Hilbert space H. We have to show that M⊥⊥=M. We can write…

Q: Find an orthonormal basis for the subspace Span ([1, 2, 0, -2], [4,3,0, 2]) of R4.

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Q: Let ū be some fixed vector in R" and let S be the set of all vectors in R" that are orthogonal to ū.…

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Q: Find a basis for the subspace of R3 spanned by S. S = {(1, 3, 6), (-1, 4, 6), (2, 4, 1)} EEE

A: Basis of subspace of ℝ3 i) It must be linearly independent. ii) Its span ℝ3

Q: Let V be the vector space of symmetric 2 × 2 matrices and W be the subspace -4 -3 -3 5 W = span{ |}.…

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Q: 2- Let H be a Hilbert space. Prove that for any two subspaces M,N of H we have (M+ N)+ M'ON. OTON…

A: Let , H be a Hilbert space. We have , M  and  N are two subspaces of H. We need to prove that , M +…

Q: Show that Null Space of Ax = 0 is a subspace. To get full credit you need to show that 1.0 is in the…

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Q: Find the orthogonal projection of onto the subspace W of R4 spanned by projw() = --B 7 010

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Q: You can apply the two-step subspace test to prove that the set of all solutions (functions)…

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Q: Let U_1, ..., U_m be subspaces of some vector space V. Show that U_1 + ... + U_m is the smallest…

A: ●Additive identities:0∈ U1 ,......,0∈ Um . ●Closed under addition: u1 ,w1 ∈ U1 ,....,um  ,wm∈ Um.…

Q: In the vector space of all real valued functions, find O (1+cos 2t, cost sint, cos² t) O (sin 2t, 1+…

A: Given spanning set is sin 2t, 1 + cos 2t, cos t sin t, cos2 t

Q: Find the coordinate vector of v = with respect to the basis -000 1 for the subspace of R4 spanned by…

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Q: (1): All vectors of the form (a,0,0) that is S={(a,0,0):u = R} Show that the following sct is…

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Q: a Is the set of vectors of the form La+2b] subspace of R3? Show your proof. a

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Q: The subspace W={( 0 -b ); a, b E R} has the basis set : —а а Select one: 0 0 a. S={(,), 1 0 0 0 0 O…

A: For a set to be a basis of a subspace. i) it must be independent. ii) it should generate the…