4. Let T: R2 → R² be defined by T = {B.[4]). and B₁ = -x y X- ([])=[ + ][²] = {[1].8]} BVB RUE BUE TOCHR be two ordered bases for R2 T (v) [X112 i) Find [7] B ii) Find [7] B and B2 = Ran B₂ [21
4. Let T: R2 → R² be defined by T = {B.[4]). and B₁ = -x y X- ([])=[ + ][²] = {[1].8]} BVB RUE BUE TOCHR be two ordered bases for R2 T (v) [X112 i) Find [7] B ii) Find [7] B and B2 = Ran B₂ [21
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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![**Problem Statement: Transformation and Bases in \(\mathbb{R}^2\)**
Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be defined by
\[ T = \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -x + y \\ x - y \end{pmatrix} \]
Additionally, consider the ordered bases:
\[ B_1 = \left\{ \begin{bmatrix} 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ -2 \end{bmatrix} \right\} \]
\[ B_2 = \left\{ \begin{bmatrix} -1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \end{bmatrix} \right\} \]
These are bases for \(\mathbb{R}^2\).
**Tasks:**
i) Find the matrix \([T]_{B_2}^{B_1}\).
ii) Find the matrix \([T]_{B_1}^{B_2}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b4adcba-fb37-46fc-90a1-02680cb6583d%2F1d381ddb-ffb4-4f4c-badf-7856c237e978%2Ffrshf3e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Transformation and Bases in \(\mathbb{R}^2\)**
Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be defined by
\[ T = \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -x + y \\ x - y \end{pmatrix} \]
Additionally, consider the ordered bases:
\[ B_1 = \left\{ \begin{bmatrix} 2 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ -2 \end{bmatrix} \right\} \]
\[ B_2 = \left\{ \begin{bmatrix} -1 \\ 2 \end{bmatrix}, \begin{bmatrix} 3 \\ 0 \end{bmatrix} \right\} \]
These are bases for \(\mathbb{R}^2\).
**Tasks:**
i) Find the matrix \([T]_{B_2}^{B_1}\).
ii) Find the matrix \([T]_{B_1}^{B_2}\).
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