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jumping from this question How to find the linear transformation associated with a given matrix?

For a given matrix $A$, let $T(v)=Av$ a linear tranformation, then is $A$ the matrix of $T$ under standard basis? i don't quite understand why. (I know it's definitely something silly) maybe because of the definition of matrix multiplication?

and if this is true, how to find a linear transformation whose matrix can also be $A$ but under not-standard basis?

i wish you could please give me an example, thank you very much!

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A Matrix is the representation of a linear transformation under a given basis. If you have a linear transformation T with matrix A under a given basis, then you can find the matrix B which represent the same linear transformation T under a different basis with the change of basis matrices:

$B= P^{-1}AQ$

Where P and Q are the change of basis matrices. You can find more information on the Wikipedia link I sent on how to find P and Q.

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    $\begingroup$ It should be $B=P^{-1}AP$ in my opinion $\endgroup$ Commented Dec 24, 2024 at 19:06
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    $\begingroup$ Only if your linear transformation does not change the vector space (endomorphism), but I agree in most cases that I have used the change of basis formula I used the less general formula that you gave. $\endgroup$ Commented Dec 24, 2024 at 19:22
  • $\begingroup$ Ohh I didn't know about that...nice $\endgroup$ Commented Dec 24, 2024 at 21:35

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