(Linear algebra)Pls answer question asapWrite work neatly and box final answersThanks8. (This is a file upload question. Be sure to show work)Let \( B = \{(1, 2), (-1, -1)\} \) and \( B' = \{(-4, 1), (0, 2)\} \) be bases for \( \mathbb{R}^2 \), and let \[ A = \begin{bmatrix} 2 & -1 \\ 0 & 1 \end{bmatrix} \]be the matrix for \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) relative to \( B \).(a) Find the transition matrix \( P \) from \( B' \) to \( B \).(b) Use the matrices \( P \) and \( A \) to find \([v]_B\) and \([T(v)]_B\), where \([v]_{B'} = \begin{bmatrix} 1 \\ -3 \end{bmatrix}\).(c) Find \( P^{-1} \).(d) Find \( A' \) (the matrix for \( T \) relative to \( B' \)).(e) Find \([T(v)]_B\) in two ways:\([T(v)]_B = P^{-1}[T(v)]_B\)\([T(v)]_B = A'[v]_{B'}\)

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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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(Linear algebra)

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Write work neatly and box final answers

Thanks

(This is a file upload question. Be sure to show work)

Let \( B = \{(1, 2), (-1, -1)\} \) and \( B' = \{(-4, 1), (0, 2)\} \) be bases for \( \mathbb{R}^2 \), and let

\[ A = \begin{bmatrix} 2 & -1 \\ 0 & 1 \end{bmatrix} \]

be the matrix for \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) relative to \( B \).

(a) Find the transition matrix \( P \) from \( B' \) to \( B \).

(b) Use the matrices \( P \) and \( A \) to find \([v]_B\) and \([T(v)]_B\), where

\([v]_{B'} = \begin{bmatrix} 1 \\ -3 \end{bmatrix}\).

(c) Find \( P^{-1} \).

(d) Find \( A' \) (the matrix for \( T \) relative to \( B' \)).

(e) Find \([T(v)]_B\) in two ways:

\([T(v)]_B = P^{-1}[T(v)]_B\)

\([T(v)]_B = A'[v]_{B'}\)

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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