Does matrix transform the $X$-$Y$ space of the vector, so it's not that the output of calculations is the vector with other data in the same basis vector space?
Let's consider the matrix:
$$ \begin{pmatrix} 2 & 3 \\ 10 & 1 \end{pmatrix} $$ multiplied by a vector $(a,b)$ to get $(8,13)$.
Does it change the basis for output or we just get the new output vector in the same basis?
Your output vector is expressed in the same basis, so if your basis was $$ e_1 = (1,0),\qquad e_2=(0,1) $$ then $$ (8,13) = 8e_1 + 13 e_2. $$