Linear Algebra Done Right · Problem 3C.4

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · matrices-linear-maps

Problem

Suppose that

D \in \mathcal{L}(\mathcal{P}_3(\mathbf{R}), \mathcal{P}_2(\mathbf{R}))

is the differentiation map defined by

Dp = p'

. Find a basis of

\mathcal{P}_3(\mathbf{R})

and a basis of

\mathcal{P}_2(\mathbf{R})

such that the matrix of

D

with respect to these bases is

\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix}.

Compare with Example 3.33. The next exercise generalizes this exercise.