In P2₁ find the change-of-coordinates matrix from the basis B = { 1 + 5t²,3 +t+16t²,1 - 6t} to the standard basis. Then write t² as a linear combination of the polynomials in B. In P2, find the change-of-coordinates matrix from the basis B to the standard basis. P = C+B (Simplify your answer.)
In P2₁ find the change-of-coordinates matrix from the basis B = { 1 + 5t²,3 +t+16t²,1 - 6t} to the standard basis. Then write t² as a linear combination of the polynomials in B. In P2, find the change-of-coordinates matrix from the basis B to the standard basis. P = C+B (Simplify your answer.)
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
Hello there, can you help me solve problem? Thank you!
![### Problem Description
In \( \mathbb{P}_2 \), find the change-of-coordinates matrix from the basis \( B = \{1 + 5t^2, 3 + t + 16t^2, 1 - 6t\} \) to the standard basis. Then write \( t^2 \) as a linear combination of the polynomials in \( B \).
---
### Task Description
In \( \mathbb{P}_2 \), find the change-of-coordinates matrix from the basis \( B \) to the standard basis.
\[
P_{\text{C} \leftarrow \text{B}} = \boxed{\phantom{insert answer here}}
\]
(Simplify your answer.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe00cebfb-aec4-474d-8979-79a2d105b819%2Fbd18c921-3958-46a6-af60-74d8f05bca81%2Fdxcdexb_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Description
In \( \mathbb{P}_2 \), find the change-of-coordinates matrix from the basis \( B = \{1 + 5t^2, 3 + t + 16t^2, 1 - 6t\} \) to the standard basis. Then write \( t^2 \) as a linear combination of the polynomials in \( B \).
---
### Task Description
In \( \mathbb{P}_2 \), find the change-of-coordinates matrix from the basis \( B \) to the standard basis.
\[
P_{\text{C} \leftarrow \text{B}} = \boxed{\phantom{insert answer here}}
\]
(Simplify your answer.)
![In \( \mathbb{P}_2 \), find the change-of-coordinates matrix from the basis \( B = \{ 1 - 2t + t^2, 4 - 7t + 5t^2, 3 - 4t + 6t^2 \} \) to the standard basis \( C = \{ 1, t, t^2 \} \). Then find the B-coordinate vector for \( 1 - 2t + 2t^2 \).
---
In \( \mathbb{P}_2 \), find the change-of-coordinates matrix from the basis \( B = \{ 1 - 2t + t^2, 4 - 7t + 5t^2, 3 - 4t + 6t^2 \} \) to the standard basis \( C = \{ 1, t, t^2 \} \).
\[
P_{C \leftarrow B} = \begin{bmatrix}
1 & 4 & 3 \\
-2 & -7 & -4 \\
1 & 5 & 6
\end{bmatrix}
\]
(Simplify your answer.)
Find the B-coordinate vector for \( 1 - 2t + 2t^2 \).
\[
[x]_B = \boxed{\phantom{entry}}
\]
(Simplify your answer.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe00cebfb-aec4-474d-8979-79a2d105b819%2Fbd18c921-3958-46a6-af60-74d8f05bca81%2Frhia5bs_processed.png&w=3840&q=75)
Transcribed Image Text:In \( \mathbb{P}_2 \), find the change-of-coordinates matrix from the basis \( B = \{ 1 - 2t + t^2, 4 - 7t + 5t^2, 3 - 4t + 6t^2 \} \) to the standard basis \( C = \{ 1, t, t^2 \} \). Then find the B-coordinate vector for \( 1 - 2t + 2t^2 \).
---
In \( \mathbb{P}_2 \), find the change-of-coordinates matrix from the basis \( B = \{ 1 - 2t + t^2, 4 - 7t + 5t^2, 3 - 4t + 6t^2 \} \) to the standard basis \( C = \{ 1, t, t^2 \} \).
\[
P_{C \leftarrow B} = \begin{bmatrix}
1 & 4 & 3 \\
-2 & -7 & -4 \\
1 & 5 & 6
\end{bmatrix}
\]
(Simplify your answer.)
Find the B-coordinate vector for \( 1 - 2t + 2t^2 \).
\[
[x]_B = \boxed{\phantom{entry}}
\]
(Simplify your answer.)
Expert Solution
Step 1: Description
Step by step
Solved in 4 steps with 4 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
