1. Let K be a n-dimension column vector space. Let A be a n x n matrix. Show that themap x EK → Ax e K" is continuous with respect to any norm || || defined on K".V Lv o V put 11/Il sellHall

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Answered: 1. Let Kn be a n-dimension column…

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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Suppose V is a finite dimensional vector space and W₁, W₂ are subspaces of V. Consider the map I: W₁ × W₂ → W₁ + W₂ defined by I'(W₁, W₂) = W₁+w₂. Prove the map is linear, and using I show that dim(W₁ + W₂) = dim(W₁) + dim(W₂) – dim(W₁ W₂)
Let P, denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: P3 → P₂ be the function that sends a polynomial to its derivative. That is, D(p(x)) = p' (x) for all polynomials p(x) E P3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + ₁x + ao and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in P3 and c E R. a. D(p(x) + q(x)) = D(p(x)) +D(q(x)) = + Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) E P3? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) E P3? choose c. Is D a linear transformation? choose (Enter a3 as a3, etc.) + +
Let Pn denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D : P3 - P2 be the function that sends a polynomial to its derivative. That is, D(p(x)) = p' (x) for all polynomials p(x) = P3. Is D a linear transformation? Let p(x) = a³x³ + ²x² + a₁ + a₁ and q(x) = b3x³ + b₂x² + b₁x + b₁ be any two polynomials in P3 and c E R. a. D(p(x) + q(x)) = . (Enter a3 as a3, etc.) D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = P3? choose b. D(cp(x)) = + c(D(p(x))) = = Does D(cp(x)) = c(D(p(x))) for all CER and all p(x) = P3? choose ✪ c. Is D a linear transformation? choose ◆
Let CE Rxn be a matrix such that xCx>0 for all nonzero vectors x ER". Prove that C is nonsingular.
20. Let f be a continuous linear functional defined on the normed linear space X. Show that M = {xe X\f(x) = 0) is a closed subspace of X.
Let denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: 03 → be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) E 3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + aº and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in 3 and c E R. a. D(p(x) + q(x)) = D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = ? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) E 3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)
Consider the vector space P, of all polynomials of degree = a, a2+b, b2 , then the magnitude of p(x)=2+1.7x equals: Answer:
1. Let h : V → W be a homomorphism. Show that the nullspace of h, (h) = { v e V | h(T) = 0w is a subspace of V.

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1. Let Kn be a n-dimension column vector space. Let A be a n x n matrix. Show that the map x E K" Ax E K" is continuous with respect to any norm || || defined on K".