Suppose that I have a vector space $V$ spanned by the columns of a matrix $V_1$, as well as by the columns of the matrix $V_2$. Suppose also that either $V_1$ and $V_2$ have a specific structure, for instance
$$V_1=[f_1(v_1) \; f_2(v_1) \cdots f_n(v_1)] \\ V_2=[f_1(v_2) \; f_2(v_2) \cdots f_n(v_2)]$$
I have found a specific linear transformation (matrix) $T$ which has a very specific structure for which
$$V_2=V_1T$$
In other words, $T$ preserves the $f_i$ function when applied to $V_1$. How can I show that I cannot find a linear transformation $T_2$ more general than my $T$ which still preserves the specific structure of $V_1$ when applied to it to create $V_2$? Does this transformation $T$ has a specific name?