1. Determine whether the following polynomials: u, v, w in P(t) are linearly dependent or independent: i. u = t³ - 4t² + 3t + 3, ii. u= t³ - 5t² - 2t + 3, v = t³ + 2t² + 4t - 1, v = t³4t² - 3t + 4, w = 2t³t² - 3t + 5; w = 2t³ 17t² - 7t + 9. 2. Write the polynomial f(t) = at² + bt + c as a linear combination of the polynomials: P₁ = (t = 1)², P2 = t - 1, P3 = 1. [Thus, P₁, P2, P3 span the space p₂ (t) of polynomials of degree ≤2]
1. Determine whether the following polynomials: u, v, w in P(t) are linearly dependent or independent: i. u = t³ - 4t² + 3t + 3, ii. u= t³ - 5t² - 2t + 3, v = t³ + 2t² + 4t - 1, v = t³4t² - 3t + 4, w = 2t³t² - 3t + 5; w = 2t³ 17t² - 7t + 9. 2. Write the polynomial f(t) = at² + bt + c as a linear combination of the polynomials: P₁ = (t = 1)², P2 = t - 1, P3 = 1. [Thus, P₁, P2, P3 span the space p₂ (t) of polynomials of degree ≤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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![1. Determine whether the following polynomials: u, v, w in P(t) are linearly dependent or
independent:
i. u = t³ - 4t² + 3t + 3,
ii. u = t³ - 5t² - 2t + 3,
v = t³ + 2t² + 4t - 1,
v = t³4t² - 3t + 4,
w = 2t³t² - 3t + 5;
w = 2t³ 17t² - 7t + 9.
2. Write the polynomial f(t) = at² + bt + c as a linear combination of the polynomials:
P₁ = (t = 1)²,
P₂ = t - 1, P3 1.
[Thus, P₁, P2, P3 span the space p₂ (t) of polynomials of degree ≤ 2]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55667b5a-7eeb-413b-818f-cce65f841660%2Fab60cfee-00b9-4cb9-8cdc-5359b98a6d6f%2Fd9x6x99_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Determine whether the following polynomials: u, v, w in P(t) are linearly dependent or
independent:
i. u = t³ - 4t² + 3t + 3,
ii. u = t³ - 5t² - 2t + 3,
v = t³ + 2t² + 4t - 1,
v = t³4t² - 3t + 4,
w = 2t³t² - 3t + 5;
w = 2t³ 17t² - 7t + 9.
2. Write the polynomial f(t) = at² + bt + c as a linear combination of the polynomials:
P₁ = (t = 1)²,
P₂ = t - 1, P3 1.
[Thus, P₁, P2, P3 span the space p₂ (t) of polynomials of degree ≤ 2]
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