Let T: P3 → R³ be defined by T (a₁ + a₁x + a₂x² + a3x³) and C= {Q.8-8} 1 Given [T Pc(T(u)). = -3 2 0 -4 2 4 0 = 3 -2 ao + 2a₁ - 3a2 + 3a3 2a02a13a2 + az .-ao − 2a₁ + a2 + 3a3. . Let u = −3x + x³, B = {x³, x², x, 1}, 1 {] 1 ‚ use the Fundamental Theorem of Matrix Representations to find ,
Let T: P3 → R³ be defined by T (a₁ + a₁x + a₂x² + a3x³) and C= {Q.8-8} 1 Given [T Pc(T(u)). = -3 2 0 -4 2 4 0 = 3 -2 ao + 2a₁ - 3a2 + 3a3 2a02a13a2 + az .-ao − 2a₁ + a2 + 3a3. . Let u = −3x + x³, B = {x³, x², x, 1}, 1 {] 1 ‚ use the Fundamental Theorem of Matrix Representations to find ,
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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![**Transformation in Vector Spaces**
Let's consider a linear transformation \( T : P_3 \to \mathbb{R}^3 \) defined by
\[
T(a_0 + a_1 x + a_2 x^2 + a_3 x^3) =
\begin{pmatrix}
a_0 + 2a_1 - 3a_2 + 3a_3 \\
2a_0 - 2a_1 - 3a_2 + a_3 \\
-a_0 - 2a_1 + a_2 + 3a_3
\end{pmatrix}
\]
Additionally, let \(\mathbf{u} = -3x + x^3\), the basis \(\mathcal{B} = \{x^3, x^2, x, 1\}\), and the basis \(\mathcal{C} = \left\{ \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \right\}\).
Given the matrix representation of the transformation \([T]_{\mathcal{C}}^{\mathcal{B}} =
\begin{pmatrix}
3 & -3 & 2 & 1 \\
-2 & 0 & -4 & 1 \\
2 & 4 & 0 & -3
\end{pmatrix}\),
we are to use the Fundamental Theorem of Matrix Representations to find \( P_{\mathcal{C}}(T(\mathbf{u})) \).
\[
P_{\mathcal{C}}(T(\mathbf{u})) =
\begin{pmatrix}
\text{Ex: 5} \\
\end{pmatrix}
\]
\[
P_{\mathcal{C}}(T(\mathbf{u})) =
\begin{pmatrix}
& \\
& \\
& \\
\end{pmatrix}
\]
The final matrix entry values for \( P_{\mathcal{C}}(T(\mathbf{u})) \) are left to be determined using the indicated theorem and given values. Review the Fundamental The](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf8aa82a-31a1-4382-a30e-3ad6a3174f84%2F158b4be3-acbc-4fbd-8e01-4c19e7304d50%2Fgfkelb_processed.png&w=3840&q=75)
Transcribed Image Text:**Transformation in Vector Spaces**
Let's consider a linear transformation \( T : P_3 \to \mathbb{R}^3 \) defined by
\[
T(a_0 + a_1 x + a_2 x^2 + a_3 x^3) =
\begin{pmatrix}
a_0 + 2a_1 - 3a_2 + 3a_3 \\
2a_0 - 2a_1 - 3a_2 + a_3 \\
-a_0 - 2a_1 + a_2 + 3a_3
\end{pmatrix}
\]
Additionally, let \(\mathbf{u} = -3x + x^3\), the basis \(\mathcal{B} = \{x^3, x^2, x, 1\}\), and the basis \(\mathcal{C} = \left\{ \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \right\}\).
Given the matrix representation of the transformation \([T]_{\mathcal{C}}^{\mathcal{B}} =
\begin{pmatrix}
3 & -3 & 2 & 1 \\
-2 & 0 & -4 & 1 \\
2 & 4 & 0 & -3
\end{pmatrix}\),
we are to use the Fundamental Theorem of Matrix Representations to find \( P_{\mathcal{C}}(T(\mathbf{u})) \).
\[
P_{\mathcal{C}}(T(\mathbf{u})) =
\begin{pmatrix}
\text{Ex: 5} \\
\end{pmatrix}
\]
\[
P_{\mathcal{C}}(T(\mathbf{u})) =
\begin{pmatrix}
& \\
& \\
& \\
\end{pmatrix}
\]
The final matrix entry values for \( P_{\mathcal{C}}(T(\mathbf{u})) \) are left to be determined using the indicated theorem and given values. Review the Fundamental The
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