What is the null space of differentiation transformation: $\frac{\mathrm{d} }{\mathrm{d} x}:P_{n} \to P_{n}$ where $P_{n}$ is the space of all polynomials of degree $\leqslant n $ over the real numbers What is the null space of the second derivative as a transformation of $P_{n}$ ? What is the null space of the kth derivative?
I am slightly at a loss here, as I realise that they are looking differentiation as a transformation, but not a simple algebraic one as that(e.g reflection over x axis or something like that, which can easily formulated into a matrix format). Can anyone :
- Give me some hints on how to approach this problem, specifically, express the differentiation as a matrix?
- Point me to some text/textbook which can help me build such concepts in a better way.
Further Edit: $P_{n}$ is as rightly pointed out is a vector space made of linear combination of the basis set {$1,x,x^{2},...,x^{n}$}. So I reckon the y=d/dx p(x) = AX where,
$p'(x)=0$ for the null space of the transformation operator.
So one trivial solution is when p(x)=c
Till now, this is what I could figure.