The set of polynomials
| f(X) = SUM_{j = 0}^{d}[ c_{j} X^{j} ] = c_{0} + c_{1} X + c_{2} X^{2} + ... + c_{d} X^{d} |
of degree d (or less if c_{d} happens to be 0) may be regarded as a vector space of dimension d + 1 by identifying a polynomial with its sequence of coefficients, i.e., the sequence (c_{0}, c_{1}, c_{2}, ..., c_{d}) which is a vector in R^{d+1}.
What formula from calculus expresses the value of c_{j} for 0 <= j <= d in terms of f ?
What rules about derivatives imply that the function D from R^{d+1} to R^{d+1} given by the operation
| f(X) ---> f'(X) |
is an abstractly linear function?
What is the kernel of D ?
What is the image of D ?
What is the matrix of D when it is expressed solely in terms of coefficients? Hint: Work it out for the special cases d = 0, 1, 2, and 3, and then surmise a pattern.