Final answer:
To compute the matrix M representing the linear transformation L (differentiation) on the polynomial space P subscript(2), apply L to the standard basis {1, x, x^2}, then express the results as linear combinations of the basis to form the columns of M. The resulting matrix M is a 3x3 matrix where the columns represent the derivatives of the basis polynomials.
Step-by-step explanation:
The student is asking about a linear transformation of polynomials, specifically how to compute the matrix representation of this transformation with respect to the standard ordered basis of the space of polynomials of degree at most 2, represented as P2. This involves finding the effect of the transformation on each basis element and expressing the result as a linear combination of the basis elements.
The standard ordered basis for P2 is typically {1, x, x2}. To compute matrix M, apply the linear transformation L to each basis vector. The outputs are then written as columns in the matrix M, using the standard basis for coefficients.
If L is the differentiation operator, L(p(x)) = p'(x), then applying L to the standard basis vectors gives:
- L(1) = 0
- L(x) = 1
- L(x2) = 2x
Expressed as linear combinations of the basis, this gives:
- L(1) = 0 * 1 + 0 * x + 0 * x2
- L(x) = 1 * 1 + 0 * x + 0 * x2
- L(x2) = 0 * 1 + 2 * x + 0 * x2
The columns of the matrix M are (0, 0, 0), (1, 0, 0), and (0, 2, 0) respectively, so matrix M is:
[0 1 0]
[0 0 2]
[0 0 0]