i'm trying to solve this problem. But simply I have no ideas.
Problem: Determine base $ B={b1,b2,b3} $ if we know that vectors $a,b,c$ are known in canonical base. Also vectors are known in base $B$. Find base $B$.
I know solution to this problem is matrix of base change.
I tried folowing:
$S:E->B$ $ $ $ $ $ $ $ $ $S^{-1}:B->S$ But im not sure how to procced. If i try to calculate it, i get horor from equations. I think there is simpler way.
The vectors $a,b,c$ are given but i didn't wrote it on purpose. I don't want someone to calculate it for me. Thank you for any tips in advance.
Suppose $S$ is the square matrix whose columns are the co-ordinates of $b_1, b_2, b_3$ relative to the canonical base. Then if $a_B$ is a column vector of co-ordinates relative to base $B$, the equivalent column vector $a_C$ of co-ordinates relative to the canonical base is given by
$a_C = Sa_B$
So if the square matrix $A_B$ is the matrix whose columns are the co-ordinates of $a,b,c$ relative to base $B$ and $A_C$ is the equivalent matrix of co-ordinates relative to the canonical base then
$A_C = SA_B$
If you are given the co-ordinates of $a,b,c$ relative to both bases then you know $A_B$ and $A_C$, so you can find $S$ from
$S=A_C(A_B)^{-1}$
Once you khave found $S$ then you can read off the co-ordinates of $b_1, b_2, b_3$ relative to the canonical base, since these are just the columns of $S$.
Note that $(A_B)^{-1}$ will only exists if $a,b,c$ are linearly independent. If $a,b,c$ are not linearly independent then you do not have enough information to find base $B$.