Let T₁ : R² → R² and T₂ : R² → R² be linear transformations defined as follows T1([2])-[-2n+bps] = T₁ T₂ I₂ = -4x1 -5x1 21₂
Let T₁ : R² → R² and T₂ : R² → R² be linear transformations defined as follows T1([2])-[-2n+bps] = T₁ T₂ I₂ = -4x1 -5x1 21₂
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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![On this educational website page, we are discussing linear transformations in the context of transformations \( T_1 \) and \( T_2 \) from \(\mathbb{R}^2\) to \(\mathbb{R}^2\) defined as follows:
\[ T_1 : \mathbb{R}^2 \to \mathbb{R}^2 \text{ and } T_2 : \mathbb{R}^2 \to \mathbb{R}^2 \]
The linear transformations are given by the following mappings:
For \( T_1 \),
\[ T_1 \left( \begin{bmatrix}
x_1 \\
x_2 \\
\end{bmatrix} \right) = \begin{bmatrix}
-4x_1 \\
-2x_1 + 5x_2 \\
\end{bmatrix} \]
For \( T_2 \),
\[ T_2 \left( \begin{bmatrix}
x_1 \\
x_2 \\
\end{bmatrix} \right) = \begin{bmatrix}
-5x_1 \\
2x_2 \\
\end{bmatrix} \]
We are asked to compute the composition of \( T_1 \) and \( T_2 \), denoted as \( (T_1 \circ T_2) \), when applied to the vector \(\begin{bmatrix}
-5 \\
0 \\
\end{bmatrix}\).
Thus, we need to find:
\[ (T_1 \circ T_2) \left( \begin{bmatrix}
-5 \\
0 \\
\end{bmatrix} \right) \]
Here, there are no additional graphs or diagrams, only the representation of the linear transformations and the operations required to find the specific composition.
Note: The expression box with "Ex: 42" is meant for the user to input their answer based on their calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf8aa82a-31a1-4382-a30e-3ad6a3174f84%2Facff4477-978c-4972-a6d7-e978db5d6dc1%2Fsic34av_processed.png&w=3840&q=75)
Transcribed Image Text:On this educational website page, we are discussing linear transformations in the context of transformations \( T_1 \) and \( T_2 \) from \(\mathbb{R}^2\) to \(\mathbb{R}^2\) defined as follows:
\[ T_1 : \mathbb{R}^2 \to \mathbb{R}^2 \text{ and } T_2 : \mathbb{R}^2 \to \mathbb{R}^2 \]
The linear transformations are given by the following mappings:
For \( T_1 \),
\[ T_1 \left( \begin{bmatrix}
x_1 \\
x_2 \\
\end{bmatrix} \right) = \begin{bmatrix}
-4x_1 \\
-2x_1 + 5x_2 \\
\end{bmatrix} \]
For \( T_2 \),
\[ T_2 \left( \begin{bmatrix}
x_1 \\
x_2 \\
\end{bmatrix} \right) = \begin{bmatrix}
-5x_1 \\
2x_2 \\
\end{bmatrix} \]
We are asked to compute the composition of \( T_1 \) and \( T_2 \), denoted as \( (T_1 \circ T_2) \), when applied to the vector \(\begin{bmatrix}
-5 \\
0 \\
\end{bmatrix}\).
Thus, we need to find:
\[ (T_1 \circ T_2) \left( \begin{bmatrix}
-5 \\
0 \\
\end{bmatrix} \right) \]
Here, there are no additional graphs or diagrams, only the representation of the linear transformations and the operations required to find the specific composition.
Note: The expression box with "Ex: 42" is meant for the user to input their answer based on their calculations.
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