B ³ = {( ¹₂ ). ( ³ ) } Tv Av where A is a matrix is a basis of R2. T: R2 R2 is a linear transformation given by - [32] 02 A =
B ³ = {( ¹₂ ). ( ³ ) } Tv Av where A is a matrix is a basis of R2. T: R2 R2 is a linear transformation given by - [32] 02 A =
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 4:**
Let \( B = \left\{ \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 5 \end{pmatrix} \right\} \) be a basis of \( \mathbb{R}^2 \). The linear transformation \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) is given by \( T\v = A\v \), where \( A \) is a matrix.
The matrix \( A \) is given by:
\[
A = \begin{bmatrix} 2 & 3 \\ 0 & 2 \end{bmatrix}
\]
Tasks:
(a) Use the definition to find \([T]^B_B\), the matrix of \( T \) under the basis \( B \).
(b) Use the definition to find \([T]^E_E\), the matrix of \( T \) under the standard basis \( E \).
(c) Compute \( P_{E \leftarrow B} \) and \( P_{B \leftarrow E} \).
(d) Use Parts (b) and (c) to find \([T]^B_B\), the matrix of \( T \) under the basis \( B \). Compare your answer with that of Part (a).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67429e19-8d15-47bc-bbfe-dfa923849540%2F639a7927-7c2f-4a06-a8a8-99bef0b0bd5a%2Fl0nl9n7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 4:**
Let \( B = \left\{ \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 5 \end{pmatrix} \right\} \) be a basis of \( \mathbb{R}^2 \). The linear transformation \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) is given by \( T\v = A\v \), where \( A \) is a matrix.
The matrix \( A \) is given by:
\[
A = \begin{bmatrix} 2 & 3 \\ 0 & 2 \end{bmatrix}
\]
Tasks:
(a) Use the definition to find \([T]^B_B\), the matrix of \( T \) under the basis \( B \).
(b) Use the definition to find \([T]^E_E\), the matrix of \( T \) under the standard basis \( E \).
(c) Compute \( P_{E \leftarrow B} \) and \( P_{B \leftarrow E} \).
(d) Use Parts (b) and (c) to find \([T]^B_B\), the matrix of \( T \) under the basis \( B \). Compare your answer with that of Part (a).
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Step 1: Define Matrix representation of T w.r.t orderd basis
VIEWStep 2: Determine the required matrix w.r.t. the basis B
VIEWStep 3: Use definition
VIEWStep 4: Determine the required matrix w.r.t. the basis E
VIEWStep 5: Determine the transition matrix from E to B
VIEWStep 6: Determine the transition matrix from B to E
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