Question

Consider the differentiation map dx/d​:P d →P d−1​ from polynomials of degree d to polynomials of degree d−1. We discussed in class how this is a surjective/onto linear transformation with a 1-dimensional null space. Express this transformation as a matrix D with respect to the standard basis B r​ ={b 0​=1,b 1​ =x,b 2​=x 2 ,…,b d​=xr } for P r​ . Describe the conditions of dx/d​ being surjective/onto and having a 1-dimensional null space in terms of pivots/free variables of D

Answer 1

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].