Determine the expressions of base change

Question

If I have the bases $\mathcal{B}_1$, $\mathcal{B}_2$, $\mathcal{B}_3$ y $\mathcal{B}_4$ of a vector space V . Known base change matrices $A = M(\mathcal{B}_1, \mathcal{B}_3)$, $B = M(\mathcal{B}_2, \mathcal{B}_3)$ y $C = M(\mathcal{B}_2, \mathcal{B}_4)$, determines what is the expression of the following base change matrices:

1. from $\mathcal{B}_2$ to $\mathcal{B}_1$.

2. from $\mathcal{B}_1$ to $\mathcal{B}_4$.

3. from $\mathcal{B}_3$ to $\mathcal{B}_4$.

Someone can help/advise me with any of the sections to be able to move forward.

Answer 1

Reading $B = M(\mathcal{B}_2, \mathcal{B}_3)$ as $B$ is the matrix that takes a vector represented according to $\mathcal{B}_2$ and represents it according to $\mathcal{B}_3$ to go the other way you invert $B$, so $B^{-1}=M(\mathcal{B}_3, \mathcal{B}_2)$. Now to go from $\mathcal{B}_2$ to $\mathcal{B}_1$ you can go from $\mathcal{B}_2$ to $\mathcal{B}_3$ and then from $\mathcal{B}_3$ to $\mathcal{B}_1$ by multiplying matrices.

Answer 2

If you want to determine a base change matrix, for example from $B_2$ to $B_1$, you have to find the coordinates of each vector of the base $B_2$ in the base $B_1$, i.e., if $B_2 = {v_1,v_2,...}$ and $B_1={w_1,w_2,...,w_n}$ you have to find the numbers $a_{ij}$ such that

$v_1 = a_{11} w_1 + a_{12}w_2+...+a_{1n} w_n$

$v_2 = a_{21}w_1 + ... + a_{2n}w_n$

... $v_n = a_{n1}w_1 + ... + a_{nn}w_n$

Then the matrix we want is formed by the coordinates of the vector, $a_{ij}$ is exactly the element of line $i$ column $j$ of the matrix we wanted to find.