In $\mathbb{R}[x]$ given linear subspace $V=span(1+x,x^2)$.
a)Find base ($f_1^*$, $f_2^*$) of space $V^*$ dual to base ($1+x$, $x^2$) of space $V$.
b)Functional $g \in V^*$ is given as $g(p)=p(0)+3p(1)$. Write $g$ as linear combination of $f_1^*$, $f_2^*$.
My solution:
a)Matrices of $f_1^*$, $f_2^*$ in base $1+x$, $x^2$ are $[1,0]$, $[0,1]$.
b)$g = f_1^*g(1+x) + f_2^*g(x^2) = 7f_1^* + 3f_2^*$
Am I right? Is it as simple as that? Do I need to write something more?
$f_1^*(a+ax+bx^2) = a$ and $f_2^*(a+ax+bx^2) = b$. Thus $$g(a+ax+bx^2)=a+3(2a+b)= 7a+3b=7f_1^*(a+ax+bx^2) + 3f_2^*(a+ax+bx^2).$$