which is the collection of all polynomials of degree ≤ 3. 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) Why is the following true? If {p1, p2, p3} spans P2 then it is a basis for P2. (1) Let p1 = 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. (5) 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) Explain why are the vectors q1 = 8 + 4x − 6x2 and q2 = −4 − 2x + 3x2 are linearly dependent in P2? (2)
which is the collection of all polynomials of degree ≤ 3. 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) Why is the following true? If {p1, p2, p3} spans P2 then it is a basis for P2. (1) Let p1 = 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. (5) 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) Explain why are the vectors q1 = 8 + 4x − 6x2 and q2 = −4 − 2x + 3x2 are linearly dependent in P2? (2)
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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which is the collection of all polynomials of degree ≤ 3.
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)
Why is the following true? If {p1, p2, p3} spans P2 then it is a basis for P2. (1)
Let p1 = 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. (5)
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)
Explain why are the vectors q1 = 8 + 4x − 6x2 and q2 = −4 − 2x + 3x2 are linearly dependent in P2? (2)
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