The matrix of Linear operator A in the basis $\{ e_1,e_2,e_3 \}$ is $$\begin{pmatrix} 1 & 2 & 0 \\ -1 & 0 & 3 \\ 2 & 1 & 5 \end{pmatrix}$$ how shall i find the matrix of this linear opeartor in the basis $\{ e_2, e_1, e_3 \}$ do i have to just change the order of lines, or is there a special algorithm to calculate the matrix in other basis?

  • $\begingroup$ You would have to change the order of columns. The easiest way to see this is to note that the columns of the matrix of $A$ are the images of the basis vectors under transformation $A$. $\endgroup$ – Vossler Mar 13 '16 at 18:31
  • $\begingroup$ You can probably figure out by asking yourself: What does that linear operator do to a vector in the given basis? And what should it do to a vector in the new basis? $\endgroup$ – Justin Benfield Mar 13 '16 at 18:32

Denote $A$ the given matrix in the basis $B=(e_1,e_2,e_3)$ and let $B'=(v_1,v_2,v_3)$ your second basis. Let $P$ the change matrix from $B$ to $B'$, then the matrix in the basis $B'$ is $P^{-1}AP$.

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    $\begingroup$ I think that, besides the general formula, it may be a good think to note that the particular case that interests the author does not require any calculations. $\endgroup$ – Vossler Mar 13 '16 at 18:36

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