4. Let\[ W = \{ p(x) \in P_3(\mathbb{R}) : p(-x) = -p(x) \}. \]\( W \) is a subspace of \( P_3(\mathbb{R}) \) (you do not need to show this). Find a basis \( \beta \) for \( W \), and prove that \( \beta \) is in fact a basis for \( W \).

Answered: Image /qna-images/answer/6fa450d0-83bc-4005-8ff2-af98f9cd4df0.jpg

Answered: 4. Let W = {p(x) € P3(R) : p(-x) =…

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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Let f(x)=-4, g(x) = -5x+7 and h(x) = -2x²-9x + 9. Consider the inner product (p,q) = p(-1)q(−1) + p(0)q(0) + p(1)q(1) in the vector space P2 of polynomials of degree at most 2. Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of P2 spanned by the polynomials f(x), g(x) and h(x). { }.
Let V = P4(R). Find a basis of V containing the polynomials p1(x) = 1 + 2x-x 2+3x 3+2x 4 , p2(x) = 2+4x+x 2+6x 3+3x 4 , p3(x) = 1+2x+2x 2+3x 3+2x 4
Exercise 1. A student is asked to prove If m is even and n is odd, then m² +n² is odd. The student provides the following proof: Proof. If m is even and n is odd, then m² + n² is odd. Since m is even, m = = 2k for any integer k. Since n is odd, n = 2k + 1. Note that m² + n² = (2k)² + (2k + 1)² = 8k² + 4k +1 So m² + n² is odd because it is an even number plus 1. (a) Identify the errors in the proof above. There are multiple errors. Explain each error. Your explanation should be written to the student who made the error and should try to help the student understand why what they wrote is incorrect. (b) In addition to errors, provide suggestions to improve the student's proof writing. That is, explain what parts of the proof are unclear or unjustified. Explain how such parts of the proof prevent some students from fully following the ment made. (c) Finally, provide a correct proof of the theorem above.
4. Consider the subspace W = 3x - 2y -2x + y x + y : x, y ER (a) Find a basis for W. (b) Find a basis for the orthogonal complement W¹. of R³.
Consider the linear mapping T: R³ R³ defined by T per 5.1. Find the matrix A of Trelative to the basis B S (D- -3+6₂ [x-y+z y-z 2x + y 5.2. Determine ker(T) and use the result to determine whether T is singular or nonsingular- тахь
State with reasons whether each of the following maps T : P2(R) → P2(R) is linear. For those maps that are linear, find their matrix with respect to the basis 1, x, x^2. (a) T(f)(x) = f(x + 1), (b) T(f)(x) = f(x) + 1.
Let F = Q(π3). Find a basis for F(π) over F.
Explain why S is not a basis for P,. S = {1 – x, 1 – x², -1 – 6x + 7x²} O sis linearly dependent. s does not span P2. S is linearly dependent and does not span P,.
2. Show all the steps clearly please
Do both parts
= [-16 12]. -9 Find bases for N(A), the null space of A, and C(A), the column space of A. Let A = Basis for N(A) is Basis for C(A) is
Let W = Span(S), where | S = 1 -2 || -2 3 | -1 | Find a basis for W and calculate dim(W).

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4. Let W = {p(x) € P3(R) : p(-x) = -p(x)}. W is a subspace of P3(R) (you do not need to show this). Find a basis B for W, and prove that B is in fact a basis for W.