In Exercises 7 and 8, find the B-matrix for the transformation x → Ax, where B = {b₁,b2}. 3 4 2 7. = · A- [_-;_ _¦], Þr = [_-¦;} Þ₂ − [2] = b₁
In Exercises 7 and 8, find the B-matrix for the transformation x → Ax, where B = {b₁,b2}. 3 4 2 7. = · A- [_-;_ _¦], Þr = [_-¦;} Þ₂ − [2] = b₁
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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![**Matrix Transformation Exercise**
In Exercises 7 and 8, find the \(\mathcal{B}\)-matrix for the transformation \( \mathbf{x} \mapsto A\mathbf{x} \), where \( \mathcal{B} = \{ \mathbf{b}_1, \mathbf{b}_2 \} \).
**Exercise 7**
\[
A = \begin{bmatrix}
3 & 4 \\
-1 & -1
\end{bmatrix}, \quad \mathbf{b}_1 = \begin{bmatrix}
2 \\
-1
\end{bmatrix}, \quad \mathbf{b}_2 = \begin{bmatrix}
1 \\
2
\end{bmatrix}
\]
In this exercise, you are given a matrix \( A \) and a basis \( \mathcal{B} \) consisting of vectors \( \mathbf{b}_1 \) and \( \mathbf{b}_2 \). Your task is to determine the \(\mathcal{B}\)-matrix representing the transformation defined by \( A \) with respect to the basis \( \mathcal{B} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F324f16c4-c4e5-4b06-b074-f261b39e021a%2Fd71f9029-91e2-4fb9-adde-fb3035f2489f%2Frh466k_processed.png&w=3840&q=75)
Transcribed Image Text:**Matrix Transformation Exercise**
In Exercises 7 and 8, find the \(\mathcal{B}\)-matrix for the transformation \( \mathbf{x} \mapsto A\mathbf{x} \), where \( \mathcal{B} = \{ \mathbf{b}_1, \mathbf{b}_2 \} \).
**Exercise 7**
\[
A = \begin{bmatrix}
3 & 4 \\
-1 & -1
\end{bmatrix}, \quad \mathbf{b}_1 = \begin{bmatrix}
2 \\
-1
\end{bmatrix}, \quad \mathbf{b}_2 = \begin{bmatrix}
1 \\
2
\end{bmatrix}
\]
In this exercise, you are given a matrix \( A \) and a basis \( \mathcal{B} \) consisting of vectors \( \mathbf{b}_1 \) and \( \mathbf{b}_2 \). Your task is to determine the \(\mathcal{B}\)-matrix representing the transformation defined by \( A \) with respect to the basis \( \mathcal{B} \).
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