I kindly request your guidance in exclusively utilizing matrix notation to solve this problem. I'm currently facing difficulties in finding a solution without employing any other methods. Could you please provide a comprehensive, step-by-step explanation using matrix notation alone to assist me in reaching the final solution?Additionally, I have provided the question and answer for reference. Could you demonstrate the matrix-based approach leading up to the solution?this has to be done the matrices way ### Problem Statement**7.15.** Let the function \( T : P_4 \to P_4 \) be defined by \( T(p(x)) = x \frac{dp}{dx} \). Show that \( T \) is a linear transformation. Find the matrix of \( T \) relative to the standard bases of \( P_4 \).---### ExplanationIn this problem, we are given a function \( T \) that maps polynomials of degree at most 4 to polynomials of degree at most 4. The transformation \( T \) is defined by \( T(p(x)) = x \frac{dp}{dx} \), where \( p(x) \) is a polynomial in \( P_4 \).1. **Linear Transformation:** - To show that \( T \) is a linear transformation, we need to demonstrate two properties for any polynomials \( p(x) \) and \( q(x) \) in \( P_4 \) and any scalar \( c \): 1. **Additivity:** \( T(p(x) + q(x)) = T(p(x)) + T(q(x)) \) 2. **Homogeneity:** \( T(c \cdot p(x)) = c \cdot T(p(x)) \)2. **Matrix Representation:** - We need to find the matrix representation of \( T \) relative to the standard basis of \( P_4 \). The standard basis for \( P_4 \) is \( \{1, x, x^2, x^3, x^4\} \). To form the matrix, we apply \( T \) to each basis element and express the result as a linear combination of the basis elements.### Detailed Steps1. **Apply \( T \) to Each Basis Element:** - \( T(1) = x \frac{d}{dx}(1) = 0 \) - \( T(x) = x \frac{d}{dx}(x) = x \cdot 1 = x \) - \( T(x^2) = x \frac{d}{dx}(x^2) = x \cdot 2x = 2x^2 \) - \( T(x^3) = x \frac{d}{dx}(x^3) = x \cdot 3x ### Diagonal Matrix Representation**7.15**\[\left[\begin{array}{cccccc}0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 \\0 & 0 & 2 & 0 & 0 \\0 & 0 & 0 & 3 & 0 \\0 & 0 & 0 & 0 & 4 \\\end{array}\right]\]This is a diagonal matrix of size \(5 \times 5\). A diagonal matrix is a type of matrix in which the entries outside the main diagonal are all zero. In this specific matrix, only the main diagonal (from the top-left to the bottom-right) contains non-zero elements, with values 1, 2, 3, and 4 respectively. ### Explanation:- **Diagonal Elements**: The numbers along the main diagonal of the matrix are 1, 2, 3, and 4.- **Non-Diagonal Elements**: All other elements in the matrix are zero. Matrices of this form are significant in linear algebra because they are simple to analyze and have straightforward properties. They are often used in eigenvalue problems and various matrix decompositions. The given numerical example, labeled as 7.15, serves as an illustration of such a matrix.

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Answered: dx 7.15. Let the function T: P4 P4 be…

dx 7.15. Let the function T: P4 P4 be defined by T (p(x)) = x. Show that T s a linear transformation. Find the matrix of T relative to the standard bases of P.

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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