Emmy Knox

2022-06-22

Consider the transformation $T:{P}_{2}\to {P}_{2}$, where ${P}_{2}$ is the space of second-degree polynomials matrices, given by $T\left(f\right)=f\left(-1\right)+f\prime \left(-1\right)\left(t+1\right)$. Find the matrix for this transformation relative to the standard basis $\mathfrak{A}=\left\{1,t,{t}^{2}\right\}$. Can someone explain to me how to find the matrix of the transformation

Korotnokby

For example: our third basis vector is ${t}^{2}$. We find that
$T\left({t}^{2}\right)=\left(-1{\right)}^{2}+2\left(-1\right)\cdot \left(t+1\right)=\left(-1\right)1+\left(-2\right)t+\left(0\right){t}^{2}$
we therefore find that the third column of our matrix is $\left(-1,-2,0\right)$. Proceed in a like fashion for the remaining columns.

Devin Anderson