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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.

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 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: $D = \begin{bmatrix}0 & 1 & 0\\0 & 0 & 2\\0 & 0 & 0\end{bmatrix}$ $D^2 = \begin{bmatrix}0 & 0 & 2\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}$ $D^3 = \begin{bmatrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}$

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