Final answer:
To express the differentiation map dx/d:P d →P d−1 as a matrix D with respect to the standard basis B r ={b₀=1,b₁=x,b₂=x²,…,bₘ=xʳ} for P r, we can consider the action of the differentiation map on each basis vector. The matrix D will have the coefficients of the resulting vectors as its entries.
Step-by-step explanation:
To express the differentiation map dx/d:P d →P d−1 as a matrix D with respect to the standard basis B r ={b₀=1,b₁=x,b₂=x²,…,bₘ=xʳ} for P r, we can consider the action of the differentiation map on each basis vector. The matrix D will have the coefficients of the resulting vectors as its entries.
For example, when applying dx/d to b₀, which is 1, we get 0, because the derivative of a constant is 0. So the first column of D will be [0, 0, 0, ..., 0].
Similarly, when applying dx/d to b₁, which is x, we get 1, because the derivative of x is 1. So the second column of D will be [1, 0, 0, ..., 0].