7. Consider the following basis B3 of P2(R) and the standard basis E of P₂ (R) B3 = {1+x², x + x²,1+2x+4x²}, E = {1, x, x²}. (a) Directly from the definition of coordinates, find [4+ 5x²] B3. (b) If [p(x)] B₂ = (c) Find the change of basis matrix PE B3. Use this matrix to solve part (b). (d) Find the change of basis matrix PB3+E. Use this matrix to solve part (a). ? find p(x).
7. Consider the following basis B3 of P2(R) and the standard basis E of P₂ (R) B3 = {1+x², x + x²,1+2x+4x²}, E = {1, x, x²}. (a) Directly from the definition of coordinates, find [4+ 5x²] B3. (b) If [p(x)] B₂ = (c) Find the change of basis matrix PE B3. Use this matrix to solve part (b). (d) Find the change of basis matrix PB3+E. Use this matrix to solve part (a). ? find p(x).
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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Question
![**Problem 7: Basis and Coordinate Transformations**
Consider the following basis \( B_3 \) of \( P_2(\mathbb{R}) \) and the standard basis \( E \) of \( P_2(\mathbb{R}) \):
\[
B_3 = \{ 1 + x^2, x + x^2, 1 + 2x + 4x^2 \}, \quad E = \{ 1, x, x^2 \}
\]
**Tasks:**
(a) Directly from the definition of coordinates, find \([ 4 + 5x^2 ]_{B_3}\).
(b) If \([ p(x) ]_{B_3} = \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix}\), find \( p(x) \).
(c) Find the change of basis matrix \( P_{E \leftarrow B_3} \). Use this matrix to solve part (b).
(d) Find the change of basis matrix \( P_{B_3 \leftarrow E} \). Use this matrix to solve part (a).
**Instructions for the Next Steps:**
For the next two questions, consider the following two bases for \( M_2(\mathbb{R}) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67429e19-8d15-47bc-bbfe-dfa923849540%2F0abfafcb-e570-47d2-b2ec-91b3cf3fb696%2Fmyrggyn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 7: Basis and Coordinate Transformations**
Consider the following basis \( B_3 \) of \( P_2(\mathbb{R}) \) and the standard basis \( E \) of \( P_2(\mathbb{R}) \):
\[
B_3 = \{ 1 + x^2, x + x^2, 1 + 2x + 4x^2 \}, \quad E = \{ 1, x, x^2 \}
\]
**Tasks:**
(a) Directly from the definition of coordinates, find \([ 4 + 5x^2 ]_{B_3}\).
(b) If \([ p(x) ]_{B_3} = \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix}\), find \( p(x) \).
(c) Find the change of basis matrix \( P_{E \leftarrow B_3} \). Use this matrix to solve part (b).
(d) Find the change of basis matrix \( P_{B_3 \leftarrow E} \). Use this matrix to solve part (a).
**Instructions for the Next Steps:**
For the next two questions, consider the following two bases for \( M_2(\mathbb{R}) \).
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