Let W be the subspace of R4 spanned byT1 = (1)Which linear functionals f(x1, x2, x3, x4) = C₁x₁ + C₂X2 + C3x3 +C4x4 are in the annihilator of W?and 7₂ =72(1).

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Answered: Let W be the subspace of R4 spanned by…

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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In the vector space P3(R), let S = {p1(x), P2(x), P3(x)}, where p1 (x) = 63x° + 19x² – 37x – 41, P2(x) = –189x – 57x² + 111x + 123, P3 (x) = 45x – 26x? – 55x + 92. (a) Determine whether S is linearly independent. (b) Determine whether (S) = P3(R). (c) Find the dimension of (S). 4.
31. Let V be the subspace of Rt defined by the equation X1 – x2 + 2x3 +4x4 = 0. Find a linear transformation T from R to R4 such that ker(T) = {0} and im(T) = V. Describe T by its matrix A.
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 ◆
Find a set of vectors that spans the subspace WCV of solutions to the following linear systems. 1. 2 *1 + 2x₂ − x3 + £₁ = 0 -3x₁ — T3 + 2x4 = 0 |T₁ + 2x3 + x₁ = 0 A
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.)
Show that T is a linear transformation

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Let W be the subspace of R4 spanned by T1 = (1) Which linear functionals f(x1, x2, x3, x4) = C₁x₁ + C₂X2 + C3x3 + C4x4 are in the annihilator of W? and 7₂ = 72 (1).