(Solved): In \( \mathbb{P}_{2} \), find the change-of-coordinates matrix from the basis \( B=\left\{1+5 t^{2 ...

In \( \mathbb{P}_{2} \), find the change-of-coordinates matrix from the basis \( B=\left\{1+5 t^{2},-4+t-19 t^{2}, 1-6 t\right\} \) to the standard basis. Then write \( t^{2} \) as a linear combination of the polynomials in \( B \). In \( \mathbb{P}_{2} \), find the change-of-coordinates matrix from the basis \( B \) to the standard basis. \[ \underset{C \leftarrow B}{P}= \] (Simplify your answer.) Write \( t^{2} \) as a linear combination of the polynomials in \( B \). \[ t^{2}=\left(1+5 t^{2}\right)+\left(-4+t-19 t^{2}\right)+(1-6 t) \]