Matrix of Differentiation Transformation D

The differentiation transformation D is a linear transformation that maps a polynomial p(x) to its derivative p'(x). In other words, D(p(x)) = p'(x). We need to find the matrix representation of D relative to the standard ordered basis of V.

Standard Ordered Basis of V

The standard ordered basis of V is {1, x, x^2, x^3}. This means that every polynomial in V can be written as a linear combination of these four basis vectors.

Matrix Representation of D

To find the matrix representation of D, we need to apply D to each of the basis vectors and express the resulting vectors as linear combinations of the basis vectors. The coefficients of these linear combinations will form the columns of the matrix representation of D.

D(1) = 0
D(x) = 1
D(x^2) = 2x
D(x^3) = 3x^2

Using these results, we can express D(1), D(x), D(x^2), and D(x^3) as linear combinations of the standard ordered basis vectors:

D(1) = 0(1) + 0(x) + 0(x^2) + 0(x^3)
D(x) = 0(1) + 1(x) + 0(x^2) + 0(x^3)
D(x^2) = 0(1) + 0(x) + 2(x^2) + 0(x^3)
D(x^3) = 0(1) + 0(x) + 0(x^2) + 3(x^3)

Therefore, the matrix representation of D relative to the standard ordered basis of V is:

| 0 0 0 0 |
| 0 1 0 0 |
| 0 0 2 0 |
| 0 0 0 3 |

This matrix can be used to apply D to any polynomial in V by multiplying the matrix by the column vector of coefficients of the polynomial.

Conclusion

In conclusion, we found the matrix representation of the differentiation transformation D relative to the standard ordered basis of the vector space V of polynomials in x over the field F of degree ≤ 3. The matrix is | 0 0 0 0 | | 0 1 0 0 | | 0 0 2 0 | | 0 0 0 3 |.