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7.16. Let the function T: P₂ ->> P3 be defined by T (p(x)) = x² dp. Show that transformation. Find the matrix of T relative to the standard bases dx T is a linear of P₂ and P3.

7.16. Let the function T: P₂ ->> P3 be defined by T (p(x)) = x² dp. Show that transformation. Find the matrix of T relative to the standard bases dx T is a linear of P₂ and P3.

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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Could you please assist me in solving both of the problem using only matrix notation? I am encountering challenges and need guidance without resorting to other methods. Can you provide a detailed explanation, step-by-step, using matrix notation exclusively to help me arrive at the solution?

Furthermore, I have attached both of the question and answer for both of the problems. Could you demonstrate the matrix approach leading to the solution?

### Section 7.16 **Problem:** Let the function \( T : P_2 \rightarrow P_3 \) be defined by \( T(p(x)) = x^2 \frac{dp}{dx} \). Show that \( T \) is a linear transformation. Find the matrix of \( T \) relative to the standard bases of \( P_2 \) and \( P_3 \). **Solution Process:** 1. **Linearity Proof:** Demonstrate that \( T \) satisfies the properties of a linear transformation: - \( T(p(x) + q(x)) = T(p(x)) + T(q(x)) \) - \( T(cp(x)) = cT(p(x)) \) for any scalar \( c \). 2. **Matrix Representation:** - Identify the standard bases for \( P_2 \) and \( P_3 \). - Apply \( T \) to each basis element of \( P_2 \) and express the result in terms of the basis elements of \( P_3 \). - Formulate the transformation matrix using these results. ### Section 7.17 **Problem:** Let the function \( T : P_2 \rightarrow P_3 \) be defined by \( T(p(x)) = \int_1^{2x} p(t) \, dt \). Show that \( T \) is a linear transformation. Find the matrix of \( T \) relative to the standard bases of \( P_2 \) and \( P_3 \). **Solution Process:** 1. **Linearity Proof:** Ensure that \( T \) meets the criteria for linear transformations: - \( T(p(x) + q(x)) = T(p(x)) + T(q(x)) \) - \( T(cp(x)) = cT(p(x)) \) for any scalar \( c \). 2. **Matrix Representation:** - Determine the standard bases for \( P_2 \) and \( P_3 \). - Compute \( T \) applied to each basis element of \( P_2 \) and write the outcomes in terms of the basis elements of \( P_3 \). - Construct the transformation matrix based on these expressions.
Transcribed Image Text:### Section 7.16 **Problem:** Let the function \( T : P_2 \rightarrow P_3 \) be defined by \( T(p(x)) = x^2 \frac{dp}{dx} \). Show that \( T \) is a linear transformation. Find the matrix of \( T \) relative to the standard bases of \( P_2 \) and \( P_3 \). **Solution Process:** 1. **Linearity Proof:** Demonstrate that \( T \) satisfies the properties of a linear transformation: - \( T(p(x) + q(x)) = T(p(x)) + T(q(x)) \) - \( T(cp(x)) = cT(p(x)) \) for any scalar \( c \). 2. **Matrix Representation:** - Identify the standard bases for \( P_2 \) and \( P_3 \). - Apply \( T \) to each basis element of \( P_2 \) and express the result in terms of the basis elements of \( P_3 \). - Formulate the transformation matrix using these results. ### Section 7.17 **Problem:** Let the function \( T : P_2 \rightarrow P_3 \) be defined by \( T(p(x)) = \int_1^{2x} p(t) \, dt \). Show that \( T \) is a linear transformation. Find the matrix of \( T \) relative to the standard bases of \( P_2 \) and \( P_3 \). **Solution Process:** 1. **Linearity Proof:** Ensure that \( T \) meets the criteria for linear transformations: - \( T(p(x) + q(x)) = T(p(x)) + T(q(x)) \) - \( T(cp(x)) = cT(p(x)) \) for any scalar \( c \). 2. **Matrix Representation:** - Determine the standard bases for \( P_2 \) and \( P_3 \). - Compute \( T \) applied to each basis element of \( P_2 \) and write the outcomes in terms of the basis elements of \( P_3 \). - Construct the transformation matrix based on these expressions.
### Matrix Representations #### Matrix Example 7.16 The matrix given in Example 7.16 is a 4x4 matrix with the following elements: \[ \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \end{bmatrix} \] This matrix is primarily filled with zeros except for a few elements located at positions (3,2), (4,3) with the values 1 and 2, respectively. The rest of the elements in the matrix are zero. #### Matrix Example 7.17 The matrix given in Example 7.17 is also a 4x4 matrix, represented as follows: \[ \begin{bmatrix} -1 & -\frac{1}{2} & -\frac{1}{3} & 0 \\ 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & \frac{8}{3} \end{bmatrix} \] This matrix includes a mixture of integers and fractional elements. Specifically: - The first row contains \(-1\), \(-\frac{1}{2}\), and \(-\frac{1}{3}\), followed by 0. - The second row contains 2, followed by three zeros. - The third row contains a 2 in the second column with the rest being zeros. - The fourth row has a single non-zero element of \(\frac{8}{3}\) in the fourth column. These matrices could be used to represent various mathematical and physical phenomena in fields such as linear algebra, physics, and engineering. Understanding these representations is crucial for solving system of equations, transformations, and other related applications.
Transcribed Image Text:### Matrix Representations #### Matrix Example 7.16 The matrix given in Example 7.16 is a 4x4 matrix with the following elements: \[ \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \end{bmatrix} \] This matrix is primarily filled with zeros except for a few elements located at positions (3,2), (4,3) with the values 1 and 2, respectively. The rest of the elements in the matrix are zero. #### Matrix Example 7.17 The matrix given in Example 7.17 is also a 4x4 matrix, represented as follows: \[ \begin{bmatrix} -1 & -\frac{1}{2} & -\frac{1}{3} & 0 \\ 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & \frac{8}{3} \end{bmatrix} \] This matrix includes a mixture of integers and fractional elements. Specifically: - The first row contains \(-1\), \(-\frac{1}{2}\), and \(-\frac{1}{3}\), followed by 0. - The second row contains 2, followed by three zeros. - The third row contains a 2 in the second column with the rest being zeros. - The fourth row has a single non-zero element of \(\frac{8}{3}\) in the fourth column. These matrices could be used to represent various mathematical and physical phenomena in fields such as linear algebra, physics, and engineering. Understanding these representations is crucial for solving system of equations, transformations, and other related applications.
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