# Thread: FInding a transition matrix

1. ## FInding a transition matrix

I know how to do this when the identity matrix is involved but not otherwise. For example can I get a step-by-step process of how this is done:

Let B = [u1, u2] and B' = [v1, v2] in R2 where
u1 = [ 2 3] u2 = [ 4 -1] v1 = [ 1 3] v2 = [-1 -1]

a) Find transition matrix from B' to B
and
b) Find transition matrix from B to B'

I know how to do this when the identity matrix is involved but not otherwise. For example can I get a step-by-step process of how this is done:

Let B = [u1, u2] and B' = [v1, v2] in R2 where
u1 = [ 2 3] u2 = [ 4 -1] v1 = [ 1 3] v2 = [-1 -1]

a) Find transition matrix from B' to B
and
b) Find transition matrix from B to B'
We need to solve the system for a,b,c,d

$v_1=au_1+bu_2 \mbox{ and } v_2=cu_1+du_2$

or in matrix form we get

$\begin{bmatrix}
2 && 4 && 1 \\
3 && -1 && 3 \end{bmatrix}$
and $\begin{bmatrix}
2 && 4 && -1 \\
3 && -1 && -1 \end{bmatrix}$

$a=\frac{13}{14},b=-\frac{3}{14},c=-\frac{5}{14},d=-\frac{1}{14}$

So the transition matrix from B' to B is

$\begin{bmatrix} a && c \\ b && d \end{bmatrix}= \begin{bmatrix} \frac{13}{14} && -\frac{3}{14} \\ -\frac{5}{14} && -\frac{1}{14} \end{bmatrix}$

To find the transition from B to B' reverse the roles of u and v or find the inverse of the matrix above.

I hope this helps. Good luck.