# Find linear map representation in another basis

А linear map has this construction in ${e_1, e_2, e_3}$ basis \begin{pmatrix}1&2&3\\-1&0&3\\2&1&5\end{pmatrix}

Find the matrix of A linear map in ${e_2,e_1,e_3}$ basis.

I have calculated the transform matrix from first basis to the second, and got \begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix} And I know that A linear map in second basis is $1/T * A * T$, where T is the transform matrix and A is the representation of linear map. (First matrix in the question). but my Transform matrix's Determinant is 0. Any suggestions?

• The determinant is $-1$ not $0$. – wgrenard Mar 29 '18 at 19:19

Observe that $T^2=I_3$, for a couple of obvious reasons.