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4. Define T: P2 → P2 by T (p) = p(0) - P(1)t + p(2)t². a. Show that T is a linear transformation. b. Find 7(p) when p(t) = −2+t. Is p an eigenvector of T? c. Find the matrix for T relative to the basis {(1,t, t²} for P₂. In Exercises 9-12. basis B for R2 with 9. A = 0 -3

4. Define T: P2 → P2 by T (p) = p(0) - P(1)t + p(2)t². a. Show that T is a linear transformation. b. Find 7(p) when p(t) = −2+t. Is p an eigenvector of T? c. Find the matrix for T relative to the basis {(1,t, t²} for P₂. In Exercises 9-12. basis B for R2 with 9. A = 0 -3

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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T:V→ V be a linear transformation with the property that T(b₁) = 7b₁ + 4b2, T(b₂) = 6b₁ – 5b₂ Find [T]B, the matrix for T relative to B. 3. Assume the mapping T: P2 → P2 defined by T (ao + a₁t + a₂t²) = 2 ao +(3 a₁ +4a2)t + (5a0-6a2)t² is linear. Find the matrix representation of T relative to the basis B = {1, t, t2}. 4. Define T: P₂ → P₂ by T (p) = p(0) - P(1)t +p(2)t². P2 a. Show that T is a linear transformation. b. Find T (p) when p(t) = −2+t. Is p an eigenvector of T? c. Find the matrix for T relative to the basis {1, t, t2} for P2. 5. Let B = {b1,b2, b3} be a basis for a vector space V. Find T(2b1 - 5b3) when T is a linear transformation from V to V whose matrix relative to B is 1 0-4 2-3 3 20 0-1 [T] B = BHA In Exercises 7 and x Ax, when B = 7. A = = [₁ 8. A = In Exercises 9-12. basis B for R2 with 9. A = =[ 10. A = 4 9 4 11. A = -1 -2 12. A = 0 -3 4 ¹=[_-1 5 -7 2 A = [1 -1
Transcribed Image Text:T:V→ V be a linear transformation with the property that T(b₁) = 7b₁ + 4b2, T(b₂) = 6b₁ – 5b₂ Find [T]B, the matrix for T relative to B. 3. Assume the mapping T: P2 → P2 defined by T (ao + a₁t + a₂t²) = 2 ao +(3 a₁ +4a2)t + (5a0-6a2)t² is linear. Find the matrix representation of T relative to the basis B = {1, t, t2}. 4. Define T: P₂ → P₂ by T (p) = p(0) - P(1)t +p(2)t². P2 a. Show that T is a linear transformation. b. Find T (p) when p(t) = −2+t. Is p an eigenvector of T? c. Find the matrix for T relative to the basis {1, t, t2} for P2. 5. Let B = {b1,b2, b3} be a basis for a vector space V. Find T(2b1 - 5b3) when T is a linear transformation from V to V whose matrix relative to B is 1 0-4 2-3 3 20 0-1 [T] B = BHA In Exercises 7 and x Ax, when B = 7. A = = [₁ 8. A = In Exercises 9-12. basis B for R2 with 9. A = =[ 10. A = 4 9 4 11. A = -1 -2 12. A = 0 -3 4 ¹=[_-1 5 -7 2 A = [1 -1
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