Exercise 18

For each vector space of functions of one real variable, represent the derivative transformation with respect to $B,B$.

• $\{\hskip 1.00006pta\cos x+b\sin x\mid a,b\in\mathbb{R}\hskip 1.00006pt\}$, $B=\langle\cos x,\sin x\rangle$

• $\{\hskip 1.00006ptae^{x}+be^{2x}\mid a,b\in\mathbb{R}\hskip 1.00006pt\}$, $B=\langle e^{x},e^{2x}\rangle$

• $\{\hskip 1.00006pta+bx+ce^{x}+dxe^{x}\mid a,b,c,d\in\mathbb{R}\hskip 1.00006pt\}$, $B=\langle 1,x,e^{x},xe^{x}\rangle$