Being trying to solve this problem on vector space but don't know how to begin with it. Tried using my knowledge on vector space but its not working because I don't know much about vector space polynomials. Express each of the following polynomials as a linear combination of the basis $2-x^2$, $x^3-x$, $2-3x^2$ and $3-x^3$, and the polynomials are:
a. $\;x^2$
b. $\;1+x$
c. $\;x^3\,$
d. $\;x+x^2$
Any idea about how to solve this problem please?
Hint: by simple elimination, let $p=2-x^2, q = x^3-x, r = 2-3x^2, s = 3-x^3\,$, then:
$$ \begin{cases} \begin{align} 3p-r &= 4 \\ q+s &= - x+3 \\ p-r &= 2 x^2 \\ s &= -x^3 + 3 \end{align} \end{cases} \quad\iff\quad \begin{cases} 1 &= \frac{1}{4}(3p-r) \\ x \;\;= 3-q-s &= \frac{3}{4}(3p-r)-q-s \\ x^2 &= \frac{1}{2}(p-r)\\ x^3 \;\;= 3 - s &= \frac{3}{4}(3p-r) - s \end{cases} $$
Given any polynomial $\,f(x)=a\cdot x^3+b\cdot x^2+c\cdot x+ d \cdot 1\,$ in the canonical base, replacing $\,1, x, x^2, x^3\,$ with the expressions above will give $\,f\,$ as a linear combination of $\,p,q,r,s\,$.