I am having trouble finding the transition matrix between two bases, $A$ and $B$.

$A = \begin{bmatrix}{t+1, t-1}\end{bmatrix}$ and $B = \begin{bmatrix}{2t+1, 3t+1}\end{bmatrix}$.

I understand that I want to find a matrix to multiply $A$ by to get $B$, but do not see how to get from $t+1$ to $2t+1$...

I know that my answer must be a matrix with maximum $2$ columns (multiplying matrices to get an output of $B$). I've tried multiplying by a $2x2$ matrix but cannot figure out what to put in the first column to get from $t+1$ to $2t+1$


I would start off by working out what linear combinations of the elements in basis $A$ is needed to equal the elements of basis $B$. There are many ways to do this, but if it isn't immediately obvious, it is useful to break things down as follows:

$\frac{1}{2}\big((t+1)+(t-1)\big) = t$

$\frac{1}{2}\big((t+1)-(t-1)\big) = 1$

So it is now clear to see that,

$2t+1 = 2\big(\frac{1}{2}\big((t+1)+(t-1)\big)\big)+\frac{1}{2}\big((t+1)-(t-1)\big) = \frac{3}{2}(t+1) + \frac{1}{2}(t-1)$

Likewise, you can work out what coefficients of $(t+1)$ and $(t-1)$ are required to equal $3t+1$.

This should help in finding the transition matrix (by using the coefficients as entries).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.