# Finding the inverse of linear transformation using matrix. A linear transformation represented by a matrix with respect to some random bases, how could I find the inverse of the transformation using the matrix representation?

Finding the inverse of linear transformation using matrix. A linear transformation represented by a matrix with respect to some random bases, how could I find the inverse of the transformation using the matrix representation?
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Let $T$ be an invertible linear transformation from an $n$-dimensional vector space to another, $A$ its $\left(n×n\right)$ matrix, $\mathcal{B}=\left\{{\alpha }_{1},\dots ,{\alpha }_{n}\right\}$ a basis for the domain, ${\mathcal{B}}^{\prime }=\left\{{\beta }_{1},\dots ,{\beta }_{n}\right\}$ a basis for the co-domain, and $\beta ={y}_{1}{\beta }_{1}+\cdots +{y}_{n}{\beta }_{n}$ a vector in the co-domain.
The inverse of $T,{T}^{-1},$, is related to ${A}^{-1}$ by ${\left[{T}^{-1}\beta \right]}_{\mathcal{B}}={A}^{-1}\left[\beta {\right]}_{{\mathcal{B}}^{\prime }},$, where $\left[\cdot {\right]}_{\mathcal{B}}$ denotes the coordinate matrix relative to the ordered basis $\mathcal{B}.$ Then with ${A}_{ij}^{-1}$ representing the $ij$ entry of ${A}^{-1},$, we compute
$\begin{array}{rl}{\left[{T}^{-1}\beta \right]}_{\mathcal{B}}& ={A}^{-1}\left[\beta {\right]}_{{\mathcal{B}}^{\prime }}\\ \\ & ={A}^{-1}\left[{y}_{1}{\beta }_{1}+\cdots +{y}_{n}{\beta }_{n}{\right]}_{{\mathcal{B}}^{\prime }}\\ \\ & ={A}^{-1}\left[\begin{array}{c}{y}_{1}\\ ⋮\\ {y}_{n}\end{array}\right]\\ \\ & =\left[\begin{array}{c}\sum _{i=1}^{n}{A}_{1i}^{-1}{y}_{i}\\ ⋮\\ \sum _{i=1}^{n}{A}_{ni}^{-1}{y}_{i}\end{array}\right];\\ \\ {T}^{-1}\beta & ={\alpha }_{1}\sum _{i=1}^{n}{A}_{1i}^{-1}{y}_{i}+\cdots +{\alpha }_{n}\sum _{i=1}^{n}{A}_{ni}^{-1}{y}_{i}.\end{array}$
If you are satisfied having your inverse in terms of $\mathcal{B},$, we are done.
If, however, you want the inverse in terms of the standard ordered basis, express each ${\alpha }_{i}$ in terms of the standard basis, and group the terms in the last expression above by the elements of the standard basis.