P2 find the change-of-coordinates matrices from basis B to basis C. {1−2t+t², 4+7t + 5t², 5-8t+8t²), C={1, t, t²} [1-21] [ 1 45] [145] -16 -7 3
P2 find the change-of-coordinates matrices from basis B to basis C. {1−2t+t², 4+7t + 5t², 5-8t+8t²), C={1, t, t²} [1-21] [ 1 45] [145] -16 -7 3
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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![### Change-of-Coordinates Matrices in Polynomial Spaces
**Task:**
In \( \mathbb{P}_2 \), find the change-of-coordinates matrices from basis \( B \) to basis \( C \).
- Basis \( B = \{1 - 2t + t^2, \; 4 + 7t + 5t^2, \; 5 - 8t + 8t^2\} \)
- Basis \( C = \{1, \; t, \; t^2\} \)
**Options for the change-of-coordinates matrix:**
**Option a:**
\[ \begin{bmatrix}
1 & -2 & 1 \\
4 & -7 & 5 \\
5 & -8 & 8
\end{bmatrix} \]
**Option b:**
\[ \begin{bmatrix}
1 & 4 & 5 \\
-2 & -7 & -8 \\
1 & 5 & 8
\end{bmatrix} \]
**Option c:**
\[ \begin{bmatrix}
1 & 4 & 5 \\
-2 & 7 & -8 \\
1 & 5 & 8
\end{bmatrix} \]
**Option d:**
\[ \begin{bmatrix}
-16 & -7 & 3 \\
8 & 3 & -2 \\
-3 & -1 & 1
\end{bmatrix} \]
In this task, you are asked to determine which of these matrices represents the transformation from basis \( B \) to basis \( C \). The correct change-of-coordinates matrix will allow vectors expressed in the basis \( B \) to be converted accurately to vectors expressed in the basis \( C \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5483afab-7850-40f6-959f-e698dea1419a%2F22650554-47b6-4948-9865-6410bdff98e9%2Fqgdn5kd_processed.png&w=3840&q=75)
Transcribed Image Text:### Change-of-Coordinates Matrices in Polynomial Spaces
**Task:**
In \( \mathbb{P}_2 \), find the change-of-coordinates matrices from basis \( B \) to basis \( C \).
- Basis \( B = \{1 - 2t + t^2, \; 4 + 7t + 5t^2, \; 5 - 8t + 8t^2\} \)
- Basis \( C = \{1, \; t, \; t^2\} \)
**Options for the change-of-coordinates matrix:**
**Option a:**
\[ \begin{bmatrix}
1 & -2 & 1 \\
4 & -7 & 5 \\
5 & -8 & 8
\end{bmatrix} \]
**Option b:**
\[ \begin{bmatrix}
1 & 4 & 5 \\
-2 & -7 & -8 \\
1 & 5 & 8
\end{bmatrix} \]
**Option c:**
\[ \begin{bmatrix}
1 & 4 & 5 \\
-2 & 7 & -8 \\
1 & 5 & 8
\end{bmatrix} \]
**Option d:**
\[ \begin{bmatrix}
-16 & -7 & 3 \\
8 & 3 & -2 \\
-3 & -1 & 1
\end{bmatrix} \]
In this task, you are asked to determine which of these matrices represents the transformation from basis \( B \) to basis \( C \). The correct change-of-coordinates matrix will allow vectors expressed in the basis \( B \) to be converted accurately to vectors expressed in the basis \( C \).
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