its) Let P3 = {ao + a₁x + a2x² + a3x³|ao, a1, a2, a3 € R} be the vector space ofpolynomials of degrees at most 3 with real coefficients. Is{2, (5-x), (6+ x)², (8 - x)³}a basis of P3? Justify your answer.

Answered: Image /qna-images/answer/4aede6b7-de2e-442d-9423-1bf3cea60dd4.jpg

Answered: its) Let P3:= {ao+ a₁x + a2x² +…

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 P2 be the vector space of all polynomials of degree ≤ 2 with coefficients in R, andS= {1 + 2x, 1 + 3x + 5x^2 , 4x + 5x^2 } .Is Span(S)= P2?
Show that there do not exist scalars C₁, C₂, and C3 such that c₁(1,0, 1,0) + c₂ (1, 0, 2, 1) + c3(2,0, 1, 2) = (1, -2, 2, 3)
Determine if the set {(1,1, −1), (1, −1,1), (0,0,1)} is a basis of ℝ^3
Extend this to a basis of R^4
correct answer, please
Is the set of rational numbers with the usual definitions and scalar multiplication a vector space? Please show justification for answer and explain steps.
Confused. Please help:)
1) Let W: = 3a +10b+ 7c+32d 2a + 8b+ 6c +24d -5a + (-13) b + (-8) c + (-46) d] la + 2b + 1c + 8d la + 3b + 2c + 10d Then a basis for W would be: a b - 2) Let W = C : -6a-6b-5c+ d = 0 and 9a+10b+9c+ (-2) d + e = 0. d e Then a basis for W would be:
Show that the set of polynomials of degree 3 form a vector space. P(x) = ax³ + bx² + cx+d, a, b, c, d ER
Find a basis for the column space of 4 1 3 0 - -31 3 2 1 4 -1 A -3 -2 -1 - -4 2 6 -4 -2 -8 3 Basis }. ={ 9.
Show your work with good explanation.

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its) Let P3:= {ao+ a₁x + a2x² + a3x³|ao, a1, a2, a3 € R} be the vector space of polynomials of degrees at most 3 with real coefficients. Is {2, (5-x), (6+ x)², (8 - x)³} a basis of P3? Justify your answer.