6. Prove that ||4, 14., |4, are matrix norms.Prothogonmatrix multiplicationand Q R be.orthogonal, then O = ometrem

This is the problem of functional analysis.

Answered: 6. Prove that ||4| , 14. , |4. are…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Be v,.,V,w vectors in R" It is known that there is a single solution to the equation x,v, +…+x,v½ = w_ Prove that the group {v,,v½} Linear independent
Which of the following sets forms a basis for R³? (:) ()} )( 8 2 -6 А. -7 3 10 -15 -8 7 3 В. -9 -8 -17 8 14 -3 -3 -2 -7 C. 9 –6 -9 1 9. » {(;) (}) ()} * {(;) ()} -5 7 6. D. -4 7 4 9 -10 6 4 Е. -2 1 4
What are all the 2 by 2 matrices of O's and l's that are invertible ( mod 2) ?
2. Determine if the function (u, v) = U₁V₂ + U₂V1, where u = each axiom to determine if it holds or fails. Show all work. (u₁, ₂) and 7 = (V₁, V₂) defines an inner product on R². Check
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3. Let x ER", A, B E Rnxn and || || be some vector norm (same notation is used for the corresponding operator norm). Show that || Ax|| ≤ ||A||||||, ||AB|| ≤ ||A|||| B|| and ||I|| = 1. Show by counterexample that ||- ||max and ||- ||F are not operator norms. ||AB|| 2 Hint: Find matrices A, B € R²×2 such that some of the properties in (1) do not hold. Note that ||I|| max = 1. In addition, the Frobenius norm satisfies the second inequality in (1): let AT = [a₁,..., an] and B = [b₁, ...,b], so that n n n n ·Σ (a{b;)² ≤ Σ ||a₁||²||6,;||³² = Σ || a;||²Σ||6; ij=1 i,j=1 i=1 j=1 n n = Σ |aki|² Σ |b₁j|² = || A||²|| B||²| i,k=1 j,l=1 (1) where the Cauchy-Schwarz inequality was used. 1/2
12. Consider the basis * {1 +x + x?, x +x², x²}. B = for P2. Let p(x) -1 + x + 2x2. Then %3D [p(x)]B = 2 None 2 1
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6. Prove that ||4| , 14. , |4. are matrix norms.