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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Problem #4
![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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27298f72-194b-4325-bcb2-0e9456bede18%2Fc4cbc43e-0e57-451b-ae94-e5bd06e63a16%2F0o4o6gi_processed.jpeg&w=3840&q=75)
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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