Assume that T is a linear transformation. Find the standard matrix of T. T: R3→R2, T (e,) = (1,9), and T (e2) = (-6,2), and T (e3) = (9,- 5), where e,, e2, and ez are the columns of the 3x3 identity matrix. ..... A = (Type an integer or decimal for each matrix element.)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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### Educational Content: Understanding Linear Transformations and Standard Matrices

#### Problem Statement:
Assume that \( T \) is a linear transformation. Find the standard matrix of \( T \).

#### Given:
\[ T: \mathbb{R}^3 \rightarrow \mathbb{R}^2 \]

- \( T(\mathbf{e_1}) = (1, 9) \)
- \( T(\mathbf{e_2}) = (-6, 2) \)
- \( T(\mathbf{e_3}) = (9, -5) \)

where \( \mathbf{e_1}, \mathbf{e_2}, \mathbf{e_3} \) are the columns of the \( 3 \times 3 \) identity matrix.

#### Task:
Find the matrix \( A = \begin{bmatrix} \quad \end{bmatrix} \)

*(Type an integer or decimal for each matrix element.)*

#### Explanation:
The transformation \( T \) maps vectors from \( \mathbb{R}^3 \) to \( \mathbb{R}^2 \). The standard matrix \( A \) representing this transformation can be constructed using the images of the standard basis vectors \( \mathbf{e_1}, \mathbf{e_2}, \mathbf{e_3} \) under \( T \).

The columns of matrix \( A \) are the images of these standard basis vectors. Therefore, the standard matrix \( A \) is constructed as follows:

\[
A = \begin{bmatrix}
1 & -6 & 9 \\
9 & 2 & -5 \\
\end{bmatrix}
\]
Transcribed Image Text:### Educational Content: Understanding Linear Transformations and Standard Matrices #### Problem Statement: Assume that \( T \) is a linear transformation. Find the standard matrix of \( T \). #### Given: \[ T: \mathbb{R}^3 \rightarrow \mathbb{R}^2 \] - \( T(\mathbf{e_1}) = (1, 9) \) - \( T(\mathbf{e_2}) = (-6, 2) \) - \( T(\mathbf{e_3}) = (9, -5) \) where \( \mathbf{e_1}, \mathbf{e_2}, \mathbf{e_3} \) are the columns of the \( 3 \times 3 \) identity matrix. #### Task: Find the matrix \( A = \begin{bmatrix} \quad \end{bmatrix} \) *(Type an integer or decimal for each matrix element.)* #### Explanation: The transformation \( T \) maps vectors from \( \mathbb{R}^3 \) to \( \mathbb{R}^2 \). The standard matrix \( A \) representing this transformation can be constructed using the images of the standard basis vectors \( \mathbf{e_1}, \mathbf{e_2}, \mathbf{e_3} \) under \( T \). The columns of matrix \( A \) are the images of these standard basis vectors. Therefore, the standard matrix \( A \) is constructed as follows: \[ A = \begin{bmatrix} 1 & -6 & 9 \\ 9 & 2 & -5 \\ \end{bmatrix} \]
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