Question

Let V be the real vector space of polynomials
over R of degree at most 2 and W be the
subspace of V generated by (1+x², 1+2x² }.
Find the kernel of the differential linear
operator d/dx on W

Answer 1

$V=\{a_{0}+a_{1}x+a_{2}x^{2}|a_{0},a_{1},a_{2}\in R\}\\ \\ W=<1+x^{2},1+2x^{2}>$

Any vector of W is of the form $a(1+x^{2})+b(1+2x^{2})$ =(a+b)+(a+2b)$x^{2}$

Let T=$\dfrac{d}{dx}$

As T operates on W, according to the definition of differential operator

T($(a+b)+(a+2b)x^{2})=2x(a+2b)$

Hence Kernel (T)=$\{p\in W|p=0\}$ where p is a polynomial of W.

$\Rightarrow T ((a+b)+(a+2b)x^{2}=0$

$\Rightarrow 2x(a+2b)=0$

$\Rightarrow (a+2b)=0$

$\Rightarrow a=-2b$

Therefore P=(-2b+b)+(-2b+2b)x$^{2}$ =-b , which is a constant polynomial.

As $b\in R\\$ , Kernel(T) is isomorphic to R.