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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 $2t+1$

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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).

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