Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Linear Transformations and Matrix Representations**
8. For each of the linear transformations below, write the matrix of the linear transformation.
a. \( T: \vec{x} \in \mathbb{R}^3 \rightarrow T(\vec{x}) \in \mathbb{R}^3 \), where \( T \) is given by:
\[
T \left( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \right) =
\begin{bmatrix}
2x_1 - 4x_2 \\
x_1 - x_3 \\
-x_2 + 3x_3 \\
3x_1 - 2x_2
\end{bmatrix}
\]
b. \( T: \vec{x} \in \mathbb{R}^2 \rightarrow T(\vec{x}) \in \mathbb{R}^3 \), where \( T \) is given by:
\[
T \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) =
\begin{bmatrix}
x_1 + 4x_2 \\
x_2
\end{bmatrix}
\]
c. Consider a polynomial in \( P_2 \) given by \( p(t) = a_0 + a_1 t + a_2 t^2 \). Define a linear operator \( T \) by \( T(p(t)) = (2t^2 - t + 6)p(t) \) in \( P_4 \). Find the matrix of the transformation. [Hint: See Example 2.]
d. Consider a polynomial in \( P_3 \) given by \( p(t) = a_0 + a_1 t + a_2 t^2 + a_3 t^3 \). Find the matrix of the linear transformation taking this vector into \( P_2 \) defined by the derivative operator \( \frac{d}{dt}[p(t)] \).
e. Consider the function defined as \( y(x) = a_1 e^x + a_2 e^{-x} + a_3 e^{5x} + a_4 e^{-7x} \). Write the matrix of the linear transformation defined by the derivative operator \( \frac](https://content.bartleby.com/qna-images/question/410816ed-4be2-4291-b0fb-4c8f4b2206ab/8879fd53-eee4-41ec-903b-1ab7d23c5967/y78i8b_processed.jpeg)
Transcribed Image Text:**Linear Transformations and Matrix Representations**
8. For each of the linear transformations below, write the matrix of the linear transformation.
a. \( T: \vec{x} \in \mathbb{R}^3 \rightarrow T(\vec{x}) \in \mathbb{R}^3 \), where \( T \) is given by:
\[
T \left( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \right) =
\begin{bmatrix}
2x_1 - 4x_2 \\
x_1 - x_3 \\
-x_2 + 3x_3 \\
3x_1 - 2x_2
\end{bmatrix}
\]
b. \( T: \vec{x} \in \mathbb{R}^2 \rightarrow T(\vec{x}) \in \mathbb{R}^3 \), where \( T \) is given by:
\[
T \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \right) =
\begin{bmatrix}
x_1 + 4x_2 \\
x_2
\end{bmatrix}
\]
c. Consider a polynomial in \( P_2 \) given by \( p(t) = a_0 + a_1 t + a_2 t^2 \). Define a linear operator \( T \) by \( T(p(t)) = (2t^2 - t + 6)p(t) \) in \( P_4 \). Find the matrix of the transformation. [Hint: See Example 2.]
d. Consider a polynomial in \( P_3 \) given by \( p(t) = a_0 + a_1 t + a_2 t^2 + a_3 t^3 \). Find the matrix of the linear transformation taking this vector into \( P_2 \) defined by the derivative operator \( \frac{d}{dt}[p(t)] \).
e. Consider the function defined as \( y(x) = a_1 e^x + a_2 e^{-x} + a_3 e^{5x} + a_4 e^{-7x} \). Write the matrix of the linear transformation defined by the derivative operator \( \frac
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