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T: R³ R², 7(e₁) = (1,3), T(e₂) = (4,2), and 7(e) = (-5, 4), where e₁,e2, e3 are the columns of the 3 x 3 identity matrix.

T: R³ R², 7(e₁) = (1,3), T(e₂) = (4,2), and 7(e) = (-5, 4), where e₁,e2, e3 are the columns of the 3 x 3 identity matrix.

Advanced Engineering Mathematics
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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Assume that T is a linear transformation. Find the standard matrix of T. 

Let \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) be a linear transformation defined by its action on the standard basis vectors. The transformation is given by: - \( T(e_1) = (1, 3) \) - \( T(e_2) = (4, 2) \) - \( T(e_3) = (-5, 4) \) where \( e_1 \), \( e_2 \), and \( e_3 \) are the columns of the \( 3 \times 3 \) identity matrix. This linear transformation maps vectors from a three-dimensional space (\( \mathbb{R}^3 \)) to a two-dimensional space (\( \mathbb{R}^2 \)). The vectors \( e_1 \), \( e_2 \), and \( e_3 \) are the standard basis vectors in \( \mathbb{R}^3 \).
Transcribed Image Text:Let \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) be a linear transformation defined by its action on the standard basis vectors. The transformation is given by: - \( T(e_1) = (1, 3) \) - \( T(e_2) = (4, 2) \) - \( T(e_3) = (-5, 4) \) where \( e_1 \), \( e_2 \), and \( e_3 \) are the columns of the \( 3 \times 3 \) identity matrix. This linear transformation maps vectors from a three-dimensional space (\( \mathbb{R}^3 \)) to a two-dimensional space (\( \mathbb{R}^2 \)). The vectors \( e_1 \), \( e_2 \), and \( e_3 \) are the standard basis vectors in \( \mathbb{R}^3 \).
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