Suppose I was given an $n\times n$ matrix $A$ that corresponds to a transformation $\alpha_A:\mathbb{R}^n\rightarrow\mathbb{R}^n$. The basis of this $A$ corresponding to $\alpha_A$ isn't given. Now I am asked to find the matrix corresponding to $\alpha_A$ with respect to another basis, say $\{v_1,v_2\}$. How do I do this?
My solution for this is as follows:
Let $B$ be the matrix corresponding to $\alpha_A$ with respect to the standard basis of $\mathbb{R}^n$. $A$ would then be $$ A=S^{-1}BS $$ such that the elements of $S$ corresponds to the basis of $A$, i.e. if $w_i$ is the basis of $A$ for $\alpha_A$, then$$ w_i=(s_{1i}, s_{2i},....s_{ni})^T $$ From here I can get the $S$ matrix and therefore get the basis $w_i$. From these I can then get the change of basis matrix from basis $w_i$ to $v_i$, and from there the matrix of $\alpha_A$ with respect to $v_i$.
But now my problem is how do I find the matrix of $\alpha_A$ with respect to the standard basis of $\mathbb{R}^n$ (i.e. the matrix $B$)?
Is this the correct approach to this problem?