Let $ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} $, $ \mathbf{v}_1 = \begin{bmatrix} -8 \\ 8 \end{bmatrix} $, and $ \mathbf{v}_2 = \begin{bmatrix} -5 \\ -3 \end{bmatrix} $.Let $ T: \mathbb{R}^2 \to \mathbb{R}^2 $ be a linear transformation that maps $ \mathbf{x} $ into $ x_1 \mathbf{v}_1 + x_2 \mathbf{v}_2 $. Find a matrix $ A $ such that $ T(\mathbf{x}) $ is $ A\mathbf{x} $ for each $ \mathbf{x} $.\[ A = \boxed{\phantom{A}} \]

Given that and .The objective is to find the matrix in which .

Answered: -8 , and V₂ = ---- Let x = X2 Ax for…

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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Question

Let $ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} $, $ \mathbf{v}_1 = \begin{bmatrix} -8 \\ 8 \end{bmatrix} $, and $ \mathbf{v}_2 = \begin{bmatrix} -5 \\ -3 \end{bmatrix} $.

Let $ T: \mathbb{R}^2 \to \mathbb{R}^2 $ be a linear transformation that maps $ \mathbf{x} $ into $ x_1 \mathbf{v}_1 + x_2 \mathbf{v}_2 $.

Find a matrix $ A $ such that $ T(\mathbf{x}) $ is $ A\mathbf{x} $ for each $ \mathbf{x} $.

\[ A = \boxed{\phantom{A}} \]

Expert Solution

Step 1: Given

Given that $x equals open square brackets table row cell x subscript 1 end cell row cell x subscript 2 end cell end table close square brackets comma v subscript 1 equals open square brackets table row cell negative 8 end cell row 8 end table close square brackets comma v subscript 2 equals open square brackets table row cell negative 5 end cell row cell negative 3 end cell end table close square brackets$ and $x subscript 1 v subscript 1 plus x subscript 2 v subscript 2$ .

The objective is to find the matrix $A$ in which $T left parenthesis x right parenthesis equals A x$ .