Exercise 10.4.4 Let T: R² R² be a linear transformation defined by →>> ¹([%])=[a+b] Consider the two bases ={[][]} B2 2-{}}-{[ - ] }· = Find the matrix MB2,B₁ of T with respect to the bases B₁ and B₂. and B₁
Exercise 10.4.4 Let T: R² R² be a linear transformation defined by →>> ¹([%])=[a+b] Consider the two bases ={[][]} B2 2-{}}-{[ - ] }· = Find the matrix MB2,B₁ of T with respect to the bases B₁ and B₂. and 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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![**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%2F98a85d3f-1221-4cd9-b5c2-c225eed9a46f%2F058e603f-1a33-403b-9552-6b8df8c85584%2Fv02j09_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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