Let f : R? → R² be the linear transformation defined by -4 f(#) : 4 Let {(-1,–2), (3, 5)}, {(1, 2), (1, 1)}, B C be two different bases for R?. Find the matrix [f1% for f relative to the basis B in the domain and C in the codomain.
Let f : R? → R² be the linear transformation defined by -4 f(#) : 4 Let {(-1,–2), (3, 5)}, {(1, 2), (1, 1)}, B C be two different bases for R?. Find the matrix [f1% for f relative to the basis B in the domain and C in the codomain.
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
![Let \( f : \mathbb{R}^2 \to \mathbb{R}^2 \) be the linear transformation defined by
\[
f(\vec{x}) =
\begin{bmatrix}
-4 & 5 \\
4 & -4
\end{bmatrix}
\vec{x}.
\]
Let
\[
\mathbb{B} = \{ \langle -1, -2 \rangle, \langle 3, 5 \rangle \},
\]
\[
\mathbb{C} = \{ \langle 1, 2 \rangle, \langle 1, 1 \rangle \},
\]
be two different bases for \(\mathbb{R}^2\). Find the matrix \([f]_{\mathbb{C}}^{\mathbb{B}}\) for \( f \) relative to the basis \(\mathbb{B}\) in the domain and \(\mathbb{C}\) in the codomain.
\[
[f]_{\mathbb{C}}^{\mathbb{B}} =
\begin{bmatrix}
\underline{\hspace{2cm}} & \underline{\hspace{2cm}} \\
\underline{\hspace{2cm}} & \underline{\hspace{2cm}}
\end{bmatrix}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5cf44c6a-d320-4ff1-8744-32b01477cb58%2F441a6029-c262-45e1-a12d-a5e9f4645071%2Fcb3t0un_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( f : \mathbb{R}^2 \to \mathbb{R}^2 \) be the linear transformation defined by
\[
f(\vec{x}) =
\begin{bmatrix}
-4 & 5 \\
4 & -4
\end{bmatrix}
\vec{x}.
\]
Let
\[
\mathbb{B} = \{ \langle -1, -2 \rangle, \langle 3, 5 \rangle \},
\]
\[
\mathbb{C} = \{ \langle 1, 2 \rangle, \langle 1, 1 \rangle \},
\]
be two different bases for \(\mathbb{R}^2\). Find the matrix \([f]_{\mathbb{C}}^{\mathbb{B}}\) for \( f \) relative to the basis \(\mathbb{B}\) in the domain and \(\mathbb{C}\) in the codomain.
\[
[f]_{\mathbb{C}}^{\mathbb{B}} =
\begin{bmatrix}
\underline{\hspace{2cm}} & \underline{\hspace{2cm}} \\
\underline{\hspace{2cm}} & \underline{\hspace{2cm}}
\end{bmatrix}
\]
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