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Question 2 In this question we are considering the vector space P2, which is the collection of all polynomials of degree < 3. (2.1) Write out the standard basis for P2? What is the dimension of P2? Is it possible for the dimension to be some other number as well? Explain. (2.2) Why is the following true? If {p1, P2, P3} spans P2 then it is a basis for P2. (2.3) Let pi = 2-x+x², p2 = 1+x, p3 = x+x². Show that S = {p1. P2, P3} spans P2. Conclude that S is a basis for P2. (2.4) Using (2.3) or otherwise, write p = 3+5x – 4x2 as a linear combination of p1, P2 and p3. Show all working. Hence find (p)s, the coordinate vector of p relative to S. (2.5) Explain why are the vectors q1 = 8 + 4x – 6x2 and q2 = -4 – 2.x + 3x2 are linearly dependent in P2?

Question 2 In this question we are considering the vector space P2, which is the collection of all polynomials of degree < 3. (2.1) Write out the standard basis for P2? What is the dimension of P2? Is it possible for the dimension to be some other number as well? Explain. (2.2) Why is the following true? If {p1, P2, P3} spans P2 then it is a basis for P2. (2.3) Let pi = 2-x+x², p2 = 1+x, p3 = x+x². Show that S = {p1. P2, P3} spans P2. Conclude that S is a basis for P2. (2.4) Using (2.3) or otherwise, write p = 3+5x – 4x2 as a linear combination of p1, P2 and p3. Show all working. Hence find (p)s, the coordinate vector of p relative to S. (2.5) Explain why are the vectors q1 = 8 + 4x – 6x2 and q2 = -4 – 2.x + 3x2 are linearly dependent in P2?

Advanced Engineering Mathematics
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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TelkomSA 26,ll 4ull 99% 10:12 AM Vodacom SA You Just now Question 2 In this question we are considering the vector space P2, which is the collection of all polynomials of degree < 3. (2.1) Write out the standard basis for P2? What is the dimension of P2? Is it possible for the dimension to be some other number as well? Explain. (2.2) Why is the following true? If {p1, P2, P3} spans P2 then it is a basis for P2. (2.3) Let pi = 2-x+x2, p2 = 1+x, p3 = x+x?. Show that S = {pı, P2, P3} spans P2. Conclude that S is a basis for P2. (2.4) Using (2.3) or otherwise, write p = 3+ 5x – 4x2 as a linear combination of Show all working. Hence find (p)s, the coordinate vector of p relative to S. P1: P2 and P3. (2.5) Explain why are the vectors q1 = 8 + 4x – 6x2 and q2 = -4 – 2x + 3x? are linearly dependent in P2?
Transcribed Image Text:TelkomSA 26,ll 4ull 99% 10:12 AM Vodacom SA You Just now Question 2 In this question we are considering the vector space P2, which is the collection of all polynomials of degree < 3. (2.1) Write out the standard basis for P2? What is the dimension of P2? Is it possible for the dimension to be some other number as well? Explain. (2.2) Why is the following true? If {p1, P2, P3} spans P2 then it is a basis for P2. (2.3) Let pi = 2-x+x2, p2 = 1+x, p3 = x+x?. Show that S = {pı, P2, P3} spans P2. Conclude that S is a basis for P2. (2.4) Using (2.3) or otherwise, write p = 3+ 5x – 4x2 as a linear combination of Show all working. Hence find (p)s, the coordinate vector of p relative to S. P1: P2 and P3. (2.5) Explain why are the vectors q1 = 8 + 4x – 6x2 and q2 = -4 – 2x + 3x? are linearly dependent in P2?
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