Exercise 10.4.4 Let T: R² R² be a linear transformation defined by T ([%])=[a+b]. Consider the two bases ={[8][-]} = {[}][-4}}· Find the matrix MB₂,B₁ of T with respect to the bases B₁ and B₂. and B1 B2
Exercise 10.4.4 Let T: R² R² be a linear transformation defined by T ([%])=[a+b]. Consider the two bases ={[8][-]} = {[}][-4}}· Find the matrix MB₂,B₁ of T with respect to the bases B₁ and B₂. and B1 B2
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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![**Exercise 10.4.4**
Let \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation defined by
\[
T \left( \begin{bmatrix} a \\ b \end{bmatrix} \right) = \begin{bmatrix} a + b \\ a - b \end{bmatrix}.
\]
Consider the two bases
\[
B_1 = \left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \end{bmatrix} \right\}
\]
and
\[
B_2 = \left\{ \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \end{bmatrix} \right\}.
\]
Find the matrix \( M_{B_2, B_1} \) of \( T \) with respect to the bases \( B_1 \) and \( B_2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F52fb4760-8b56-41d1-811f-26cd2f9d5b41%2F34552956-3d19-4667-8c38-e89b7b78de20%2Fvmm6ox_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 10.4.4**
Let \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation defined by
\[
T \left( \begin{bmatrix} a \\ b \end{bmatrix} \right) = \begin{bmatrix} a + b \\ a - b \end{bmatrix}.
\]
Consider the two bases
\[
B_1 = \left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} -1 \\ 1 \end{bmatrix} \right\}
\]
and
\[
B_2 = \left\{ \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \end{bmatrix} \right\}.
\]
Find the matrix \( M_{B_2, B_1} \) of \( T \) with respect to the bases \( B_1 \) and \( B_2 \).
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