Semaj Christian

2022-06-25

to with the conditions that $F\left({p}_{0}\right)=f$, $F\left({p}_{1}\right)=g$, $F\left({p}_{2}\right)=h$ where $f\left(x\right)={x}^{2}+3$, $g\left(x\right)={x}^{2}-x$, $h\left(x\right)=2+x$. Find the transformation matrix for $F$ in the basis $\left({p}_{0},{p}_{1},{p}_{2}\right)$

pheniankang

The problem is meaningless without the knowledge of ${p}_{0}$, ${p}_{1}$, and ${p}_{2}$. If it turns out that ${p}_{0}=1$, that ${p}_{1}=x$, and that ${p}_{2}={x}^{2}$, then, since $f\left({p}_{0}\right)=3{p}_{0}+{p}_{2}$, $f\left({p}_{1}\right)=-{p}_{1}+{p}_{2}$, and $f\left({p}_{2}\right)=2{p}_{0}+{p}_{1}$, the matrix that you're after is
$\left[\begin{array}{ccc}3& 0& 2\\ 0& -1& 1\\ 1& 1& 0\end{array}\right].$

Leland Morrow

Cheers! I can only imagine it has to be this since otherwise the exercise seems to make little sense. The exercise has a second part: "What is the image of the transformation $x↦3{x}^{2}+2x+1$ in this transformation?" Can I solve that simply by multiplying the matrix by the vector "$3{x}^{2}+2x+1$" or, I assume, "$3{p}_{2}+2{p}_{1}+{p}_{0}$"?