Question

help

Let P1 be the linear space of real polynomials of degree at most one, so a typical element
is p(x) := a + bx, where a and b are real numbers. The derivative, D : P1 → P1 is,
as you should expect, the map DP(x) = b = b + 0x. Using the basis e1(x) := 1,
e2(x) := x for P1 , we have p(x) = ae1(x) + be2(x) so Dp = be1 .
Using this basis, find the 2 × 2 matrix M for D. Note the obvious property D2p = 0
for any polynomial p of degree at most 1. Does M also satisfy M2 = 0? Why should
you have expected this?

Answer 1

That looks hard man

Answer 2

they want the matrix that gives
\[\left[\begin{matrix}? & ? \\ ? & ?\end{matrix}\right]\left[\begin{matrix}a \\ b\end{matrix}\right]=\left[\begin{matrix}b \\ 0\end{matrix}\right]\]