Chapter 3: Chapter 3

Problem 76

Let $\left\\{1, \mathrm{x}, \mathrm{x}^{2}\right\\}$ be a basis for the vector space of polynomials of degree not exceeding 2 . Let $\mathrm{D}, \mathrm{D}^{2}$ and $\mathrm{D}^{3}$ denote differentiation operators. Find the matrices of $\mathrm{D}, \mathrm{D}^{2}$ and $\mathrm{D}^{3}$ relative to the above basis.

Short Answer

Expert verified

The matrices of the differentiation operators D, D^2, and D^3 relative to the given basis are: \[D = \begin{bmatrix}0 & 1 & 0\\0 & 0 & 2\\0 & 0 & 0\end{bmatrix}\] \[D^2 = \begin{bmatrix}0 & 0 & 2\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}\] \[D^3 = \begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}\]

See the step by step solution

Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Apply Differentiation Operator to Basis Elements

First, we will apply the differentiation operator D to each element of the basis: 1. D(1) = 0 2. D(x) = 1 3. D(x^2) = 2x

Step 2: Express the Resulting Polynomials in Terms of the Basis

Now, we will express the resulting polynomials in terms of the basis elements: 1. 0 = 0 * 1 + 0 * x + 0 * x^2 2. 1 = 1 * 1 + 0 * x + 0 * x^2 3. 2x = 0 * 1 + 2 * x + 0 * x^2

Step 3: Construct the Matrix for Differentiation Operator D

The matrix representation for the differentiation operator D is obtained by placing the coefficients of the basis decomposition in columns. Therefore, the matrix for D is: \[D = \begin{bmatrix}0 & 1 & 0\\0 & 0 & 2\\0 & 0 & 0\end{bmatrix}\]

Step 4: Apply Differentiation Operator D^2 to Basis Elements

Now, we need to apply D^2 (the second derivative) to the basis elements. Since D^2 is the square of the differentiation operator, it is equivalent to applying D twice: 1. D^2(1) = D(D(1)) = D(0) = 0 2. D^2(x) = D(D(x)) = D(1) = 0 3. D^2(x^2) = D(D(x^2)) = D(2x) = 2

Step 5: Express the Resulting Polynomials in Terms of the Basis

We express the resulting polynomials from Step 4 in terms of the basis elements: 1. 0 = 0 * 1 + 0 * x + 0 * x^2 2. 0 = 0 * 1 + 0 * x + 0 * x^2 3. 2 = 2 * 1 + 0 * x + 0 * x^2

Step 6: Construct the Matrix for Differentiation Operator D^2

The matrix representation for the differentiation operator D^2 is obtained by placing the coefficients of the basis decomposition in columns. Therefore, the matrix for D^2 is: \[D^2 = \begin{bmatrix}0 & 0 & 2\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}\]

Step 7: Apply Differentiation Operator D^3 to Basis Elements

Now, we need to apply D^3 (the third derivative) to the basis elements. Since D^3 is the cube of the differentiation operator, it is equivalent to applying D three times: 1. D^3(1) = D(D^2(1)) = D(0) = 0 2. D^3(x) = D(D^2(x)) = D(0) = 0 3. D^3(x^2) = D(D^2(x^2)) = D(2) = 0

Step 8: Express the Resulting Polynomials in Terms of the Basis

We express the resulting polynomials from Step 7 in terms of the basis elements: 1. 0 = 0 * 1 + 0 * x + 0 * x^2 2. 0 = 0 * 1 + 0 * x + 0 * x^2 3. 0 = 0 * 1 + 0 * x + 0 * x^2

Step 9: Construct the Matrix for Differentiation Operator D^3

The matrix representation for the differentiation operator D^3 is obtained by placing the coefficients of the basis decomposition in columns. Therefore, the matrix for D^3 is: \[D^3 = \begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}\] So, the matrices of the differentiation operators D, D^2, and D^3 relative to the given basis are: \[D = \begin{bmatrix}0 & 1 & 0\\0 & 0 & 2\\0 & 0 & 0\end{bmatrix}\] \[D^2 = \begin{bmatrix}0 & 0 & 2\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}\] \[D^3 = \begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}\]

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Recommended explanations on Math Textbooks

Select your language

Chapter 3: Chapter 3

Short Answer

Step by step solution

Unlock all solutions

Step 1: Apply Differentiation Operator to Basis Elements

Step 2: Express the Resulting Polynomials in Terms of the Basis

Step 3: Construct the Matrix for Differentiation Operator D

Step 4: Apply Differentiation Operator D^2 to Basis Elements

Step 5: Express the Resulting Polynomials in Terms of the Basis

Step 6: Construct the Matrix for Differentiation Operator D^2

Step 7: Apply Differentiation Operator D^3 to Basis Elements

Step 8: Express the Resulting Polynomials in Terms of the Basis

Step 9: Construct the Matrix for Differentiation Operator D^3

Access millions of textbook solutions in one place

Most popular questions from this chapter

More chapters from the book ‘Linear Algebra’

Chapter 1

Chapter 2

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

Chapter 21

Chapter 22

Chapter 23

Chapter 24

Join over 22 million students in learning with our Vaia App

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Decision Maths

Probability and Statistics

Mechanics Maths

Calculus