$A$ and $B$ are matrices representing the same linear transformation $V$ is a two dimensional vector space and $\zeta:V\to V$ is a linear map. $A$ is a matrix of $\zeta$ with respect to the basis $\\{a,a'\\}$ and $B$ is its matrix with respect to the basis $\\{b,b'\\}$ If $A$ and $B$ are matrices representing the same linear transformation, does this mean that: $M(T,\\{a,a'\\}, \\{b,b'\\}) A = M(T,\\{a,a'\\}, \\{b,b'\\}) B$ Where $M$ is the matrix of transformation(T) from the first basis to the second.

You need to swap the order on one side of the $=$. If you have a vector in the $a$-basis, transforming that via $A$, and then change basis would be signified by $M_TA$, while first changing basis and then transforming via $B$ would be $BM_T$. Those are the ones that are equal, and they both express the transformed vector in the $b$-basis.