Answer
The change of coordinate matrix from the basis B to the standard basis is $\begin{bmatrix}
1&2&1\\0&1&2\\-3&-5&0
\end{bmatrix}$ and $[t^2]_B=\begin{bmatrix}
3\\-2\\1
\end{bmatrix}$.
Work Step by Step
The change of coordinate matrix from the basis B to the standard basis is $\begin{bmatrix}
1&2&1\\0&1&2\\-3&-5&0
\end{bmatrix}$ and the change of coordinate matrix from the standard basis to the basis B is $\begin{bmatrix}
10&-5&3\\-6&3&-2\\3&-1&1
\end{bmatrix}$
To write $t^2$ as a linear combination of the polynomials in B, we need to find the product $\begin{bmatrix}
1&2&1\\0&1&2\\-3&-5&0
\end{bmatrix}\begin{bmatrix}
0\\0\\1
\end{bmatrix}=\begin{bmatrix}
3\\-2\\1
\end{bmatrix}$
