Determine the vectors of components

[mathmari]

Hey!

For the polynomial vector space $\mathbb{R}[x]$ of degree $\leq 3$ we have the following three bases:
$$B_1 = \{1 - X^2 + X^3, X - X^2, 1 - X + X^2, 1 - X\} , \\
B_2 = \{1 - X^3, 1 - X^2, 1 - X, 1 + X^2 - X^3\}, \\
B_3 = \{1, X, X^2, X^3\}$$

How can we determine the following vectors of components $\mathbb{R}^4$ ?

$\Theta_{B_1}(b)$ for all $b \in B_1$

and

$\Theta_{B_3}(b)$ for all $b \in B_1$

Could you give me hint? (Wondering)

Do we use the transformation matrix? If yes, how? (Wondering)

 

[mathmari]

I have seen the following notes :

$\Theta_{B_1}(b\in B_1)=i$-th comlumn of idenity, since it shown always at itself, and $\Theta_{B_3}(b\in B_1)=i$-th column of $B_1$. Why does this hold? (Wondering)