Find the 3x4 matrix associated with the derivative transformation D:P3→P2 with respect to B = { 1, x, x2, x3} and B' = {1, x, x2}.
To find the 3x4 matrix associated with the derivative transformation, we need to determine the coefficients of the polynomials in P3 (polynomials of degree at most 3) with respect to the basis B' (consisting of the polynomials 1, x, and x^2).
The derivative transformation D takes a polynomial p(x) and returns its derivative p'(x).
Let's consider each polynomial in the basis B: 1, x, x^2, x^3, and differentiate them with respect to x.
1. For the polynomial 1, its derivative is 0 since it is a constant.
So the first column of the matrix will be [0 0 0].
2. For the polynomial x, its derivative is 1,
So the second column of the matrix will be [1 0 0].
3. For the polynomial x^2, its derivative is 2x,
So the third column of the matrix will be [0 2 0].
4. For the polynomial x^3, its derivative is 3x^2,
So the fourth column of the matrix will be [0 0 3].
Combining all the columns, the 3x4 matrix associated with the derivative transformation D:P3→P2 with respect to B and B' is:
| 0 1 0 0 |
| 0 0 2 0 |
| 0 0 0 3 |
Each column represents the coefficients of the corresponding polynomial in B' with respect to B.