Change of coordinate matrices

Question

Find the change of coordinate matrices:

Wherein B is the standard basis for P2 $$B' = (t^2+2,t+3,t^2+t+1) \\B" = (2t^2+t+1, t^2, 2t+1) \\ B= (t^2,t,1) $$ $$P_{B'B}$$ means the transformation for the standard basis to B'

$$ B' = \{\,t^2 + 2\,,\; t+3\,,\; t^2+t+1\,\}$$

Find

$$\\ P_{B'B} \\ P_{B"B'}$$

What are the steps to do this kinds of questions?

Answer 1

The $i^\text{th}$ column of $P_{B'B}$ should be the coordinate vector you get from expressing the $i^\text{th}$ basis element in $B$ in the basis $B'$.

For example the first basis element of $B$ is $t^2$ which we express as $t^2 = \frac{1}{2}(t^2 + 2) - \frac{1}{2}(t + 3) + \frac{1}{2}(t^2 + t + 1)$ in the basis $B'$. Thus the first column of $P_{B'B}$ is $$\begin{bmatrix}\frac{1}{2} \\ -\frac{1}{2} \\ \frac{1}{2}\end{bmatrix}.$$