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Let e₁= [8] and e₂= Y₁ = 3 7 and y₂ = into y₁ and maps €2 into Y2. Find the images of 4 -4 1 8 and let T: R² R² be a linear transformation that maps e₁ and X₁ X2

Let e₁= [8] and e₂= Y₁ = 3 7 and y₂ = into y₁ and maps €2 into Y2. Find the images of 4 -4 1 8 and let T: R² R² be a linear transformation that maps e₁ and X₁ X2

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Let \( e_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) and \( e_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \), \( y_1 = \begin{bmatrix} 3 \\ 7 \end{bmatrix} \), and \( y_2 = \begin{bmatrix} -1 \\ 8 \end{bmatrix} \), and let \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation that maps \( e_1 \) into \( y_1 \) and maps \( e_2 \) into \( y_2 \). Find the images of \( \begin{bmatrix} 4 \\ -4 \end{bmatrix} \) and \( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \). --- The image of \( \begin{bmatrix} 4 \\ -4 \end{bmatrix} \) is \(\quad \boxed{ }\). The image of \( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \) is \(\quad \boxed{ }\). --- Explanation: - The problem statement introduces two standard basis vectors \( e_1 \) and \( e_2 \), two vectors \( y_1 \) and \( y_2 \), and a linear transformation \( T \) defined by its action on these basis vectors. - Specifically, \( T \) maps \( e_1 \) to \( y_1 \) and \( e_2 \) to \( y_2 \). - The task is to use this linear transformation to find the images of two vectors: \( \begin{bmatrix} 4 \\ -4 \end{bmatrix} \) and an arbitrary vector \( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \). In a more detailed educational context, further steps would typically involve finding the explicit transformation matrix corresponding to \( T \) and using it to find the images of the given vectors.
Transcribed Image Text:Let \( e_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) and \( e_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \), \( y_1 = \begin{bmatrix} 3 \\ 7 \end{bmatrix} \), and \( y_2 = \begin{bmatrix} -1 \\ 8 \end{bmatrix} \), and let \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation that maps \( e_1 \) into \( y_1 \) and maps \( e_2 \) into \( y_2 \). Find the images of \( \begin{bmatrix} 4 \\ -4 \end{bmatrix} \) and \( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \). --- The image of \( \begin{bmatrix} 4 \\ -4 \end{bmatrix} \) is \(\quad \boxed{ }\). The image of \( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \) is \(\quad \boxed{ }\). --- Explanation: - The problem statement introduces two standard basis vectors \( e_1 \) and \( e_2 \), two vectors \( y_1 \) and \( y_2 \), and a linear transformation \( T \) defined by its action on these basis vectors. - Specifically, \( T \) maps \( e_1 \) to \( y_1 \) and \( e_2 \) to \( y_2 \). - The task is to use this linear transformation to find the images of two vectors: \( \begin{bmatrix} 4 \\ -4 \end{bmatrix} \) and an arbitrary vector \( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \). In a more detailed educational context, further steps would typically involve finding the explicit transformation matrix corresponding to \( T \) and using it to find the images of the given vectors.
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