Let {v1, ..., Vp} be an orthonormal set in Rn. Verify the following inequality, called Bessel's inequality, which is true for each x in Rn: ||xl² > \x•vi° + ]x•v2[² + • …·+ ]x•vp?

Q: 1. Let ƒ : N × N → N be defined as f(m, n) = 2m−¹(2n — 1). (a) Show that f is a bijection. (b) Using…

A:

Q: By using caushy Rieman prove that C. Prove thatd az az ae

A: According to the given information, it is required to use Cauchy Riemann to prove that:

Q: Assume that A and B are nonempty, bounded above, and satisfy B CACR. Show that sup B < sup A.

A: Introduction: The supremum and infimum properties are the very important properties of any subset of…

Q: Let A be an mxn Aqtrix Prove that = max |Ax/ -max E l9;

A:

Q: Prove that I(e) = dr, e> 0.

A:

Q: Еx6. IS V is finite dimensional ond f is a homomor- phisni of V into itself which is not onto prove…

A: If f is a homomorphism of V into itself, then f(0) = 0 .

Q: Prove that det(At) = det(A) for any A ∈ M2× 2(F).

A: Let us consider a 2*2 matrix A whose entry belongs to F, which is given below

Q: 2. Show that A is Hermitian if and only if (|A|) ER for all |t).

A: The given question is related with functional analysis. We have to show that…

Q: Prove that L is solvable if and only if adL C gl(L) is solvable.

A: Please find the answer in the next step. None that a group is said to solvable if its derived series…

Q: Define a mapping fn : Z → Z by fn(x) = 1 – nx for each integer n E Z. a. For which values of n is fn…

A:

Q: Let fca) satis fies Cauchy Riemann Eguation , then (f(z)" is also satis fies cauchy Riemann Eguation…

A:

Q: Let TE L(V, W), where V and W are finite-dimensional. Prove that rk(7) rk(7*).

A:

Q: Functional theory and functional analysis

A: According to the given information it is required to prove that:

Q: Find the infimum and supremum of the set X = {3 – Vztn ine N}. -

A:

Q: If E and E, are measurable , then (E,UE,) is not measurable. ylgn

A: False.

Q: 3. Use the above Lemma and Mathematical Induction to generalize the previous exercise. That is,…

A:

Q: 3. Show that the function p : F → Ę which maps o to o is a bijection from F to Fe, where and ar to…

A: Let ϕ:F1→F2 be a function such that, ϕ0=0. Also given that ϕαr=β3r, where 0≤r≤6 When r=0,…

Q: 1) Prove that Ym,n EN such thert m¢ nt = n<m using the princdple of in duction!

A:

Q: Show that the Modulus Function            f : R → R, given by f(x) = | x |, is neither one-one nor…

A: Consider, f: R → R given by f(x) = | x |Where,| x | is x, if x is positive or 0| x | is – x, if x is…

Q: For allne z, gcdin,nt)=l.

A:

Q: Let E Os 0s n, 0s zs 1, 8Vzs rS 8. Compute

A:

Q: Let x E Q'. Show that for any n EN, there exists Xn E Q such that |x- Xal <

A: Given x∈ℚ', that x is a fixed irrational number. We know that between any two real  number, there…

Q: Functional theory and functional analysis

A: Consider the provided question, Let A be an invertible operator acting on space X.

Q: Prove that if Tr(AT A) = 0, then A = O .

A: Proof:- Suppose A is a square matrix of order n, That is, A=aij For 1≤i,j≤n Now we need to prove…

Q: Prove that on average, permutations of length n have H, cycles, where Σ Hn i=1

A: We will take an arbitrary set containing n elements and observe the required permutations.

Q: Let h : N x N → N be the function h(x, y) = 3x + 2y. Use strong induction to prove that if n EN and…

A: Given that h : ℕ×ℕ→ℕ be the function h(x,y)=3x+2y. Then we have to prove by using strong induction…

Q: Prove that for all values of the constant A, eAInt = e In.

A:

Q: 3. In each of the following examples, determine (with careful arguments) the bound- ary of A in X,…

A: Given  X = R  and  A = (-1,1]

Q: Let H = {0, ±3, ±6, ±9,…}. How many distinct left cosets of H in ℤ are there?

A:

Q: Let m and n be positive integers with gcd(m,n) = 1. Prove that Z/mZ x _Z/ nZ = Z/mnZ

A: Suppose m and n are relatively prime.

Q: 3. Find the generator of X = Xo + ut + oWt, where W is a standard BM, and identify the PDE satisfied…

A:

Q: Let ECR be measurable. Prove that L'(E)n L°(E) C L*(E).

A: Given E⊆ℝ be a measurable set. Let f∈L7E∩L9E. Let λ=716, therefore:…

Q: . Let X be a r.V. such that E(X") is finite for all n = 1, 2,... Use the expansion

A: Introduction: Suppose X is a random variable. Then, the moment generating function (mgf) of X is…

Q: For p = (1, 0) and x = (x₁,x2,...,xn) ER", define ||2|p:= - ². |P Show that n jyjx|py|q for x, y…

A:

Q: .If o + S CR is bounded above, then S has a supremum

A: It is given that S is a non-empty sub-set of R and R is bounded above.

Q: 1). If r is not equal to one, show that a + ar + ar^2+ ....+ ar^n= a( r^n+1 -1)/( r-1).2).…

A:

Q: prove that Sa/Gn has f discription with M-ly ny-l Degrees tof freedom

A: Suppose that xi and yi are two random samples of size n, drawn from a normal population.  Population…

Q: find the desivalive d the functian ha)= ainfse and compute h'(2) Th o

A:

Q: Suppose f" exists and is bounded on Co,0), Prove that if bam fa)=0, then bim fGx)=0.

A:

Q: Let R be a UFD and fx) in REX) So, 0+a € IR.. Show that claf)~ ac(f) a nonzero polynamial

A:

Q: Assume that a1, a2, ..., an E R (with n > 2) such that a1+a2++an = 1. Prove that af+az++a > 1 (Use…

A: Cauchy-Swartz Lemma : For any real numbers  and  the following inequality holds  ,

Q: Let A E Mnxn(F) and B = aA, with a E F. Prove by induction that det(B) = a" det(A), for all positive…

A:

Q: 3. In each of the following examples, determine (with careful arguments) the bound- ary of A in X,…

A:

Q: Prove that : a"+ (a+ 1)° =1 +a(1+い) (mod a la+s) for all a, m,n E Z.

A:

Q: Let A ∈ Mn×n(F). Under what conditions is det(−A) = det(A)?

A: In determinant the scalar quantity is taken common from a single row or a column.

Q: Let A and P be an n x n matricies, where P is invertible. Show that det(PA(P-1))=detA.

A: To show that,

Q: Which of the following form an orthonormal set in a euclidean inner product space R 3 ? 101 A. 1. 0,…

A:

Q: Prove that for any k > 0, nk e o(2v™).

A: The complete solutions are given below

Q: Prove that det(kA) = kn det(A) for any A ∈ Mn×n(F).

A: In determinant the scalar quantity is taken common from a single row or a column.