Problem 4: Consider the set of polynomials {2+ t, 1 – 5t}. Let p¡(t) = 2 + t andP2(t) = 1 – 5t. Let B ={2+1, 1 – 5t}.(a) Show that the set B is a basis for P1. Show your work (define a basis if you use acoordinate mapping).(b) If [v (1)],find v(t).B(c) Find [1- 51(d) Find -2+ 43t]B(e) Find 0 (t) where 0 (t) = 0 + Ot.B

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Answered: Problem 4: Consider the set of…

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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Problem 3: Consider the set of polynomials {-1+3t – 21², 4 – 10t + 3t², 4 – 8t – 21²} . Let p¡(t) = – 1 + 3t – 21², p2(t) = 4 – 10t + 3ť², and p3(t) = 4 – 8t – 21². Let H = span {p,(t), P2(t), P3(t)}, which is a subset of P,.
The set B = {1– 2t + t²,3 – 5t + 4t2 , 2t + 3t²} is a basis of P2. Suppose C = {c1, c2, C3} is another basis of P2 such 4 11 that P = C+B Determine the polynomials c1(t), c2(t), and c3(t). 1 3 -1
P(x)] B The set B = {− (2 + 4x²), x −6 − 12x², 2x – 4 – 12x²} is a basis for P₂. Find the coordinates of p(x) = 20 − 3x + 40x² relative to this basis:
(b) Show that the set {V₁, V2, V3, V₁, V₁} = {1+x, 1-x, x² + x³, 1-x², x² + x²} is a basis for P4 (R).
Show that T(x, y, z) = (4x + 2y – 2z, −2x + y + 3z, x - y - 2z) is not a one-to-one transformation from R³ to R³. Find a basis of the kernel of this transformation.
= [-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
Consider the basis {1, 1 - t, 1 + t + t2} of the space P2 of polynomials of degree at most two. Find the coordinate vector of t - t2 with respect to the standard basis.
-CERKCHEFÐ (a) Find the coordinates of x with respect to basis B. (b) Find the coordinates of x with respect to basis C. (c) Find the change of coordinate matrix P. Use it to show it maps your coordinates from part (a) to your coordinates from part (b). C+B 1. Let B and x = (d) Find P by inverting your answer in part (c). Check that it maps the appropriate coordinates. B-C
et △ABC be a triangle in S^2 (the two-dimensional sphere). Let the dual point C'∈S^2 of C be defined by the following three conditions:(i) d(C' , A)=π/2(ii) d(C' , B)=π/2(iii) d(C' , C)≤π/2A' and B' are defined analogously. Thus we get a dual triangle △A' B' C'. More precisely, △ABC is a non-degenerate triangle, andit follows (you may assume) that △A' B' C' is too.(question) Are there triangles △ABC in S^2 identical to their own dual △A'B'C'? I would be very thankful if you could provide some explanation with the steps, thank you in advance.

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Problem 4: Consider the set of polynomials {2+ t, 1 – 5t}. Let p¡(t) = 2 + t and P2(t) = 1 – 5t. Let B = {2+1, 1 – 5t}. (a) Show that the set B is a basis for P1. Show your work (define a basis if you use a coordinate mapping). (b) If [v (1)], find v(t). B (c) Find [1 – 5t]