Suppose the bijective linear transformation $T_1$ maps the elements of the subspace $X$, to the subspace $X_1$. Similarly, the linear transformation $T_2$ maps elements of $X$ to $T_2$. An illustration is :
Now, my question is, can we find a transformation that takes any elements in $X_1$ or $X_2$ and map back it to $X$?
More specifically, assume for $x \in X$ , we have $T_1 x = x_1, x_1 \in X_1$ and $T_2 x = x_2, x_2 \in X_2$ . My goeal is to find a transformation $T$ such that for the mentioned $x, x_1, x_2$, $Tx_1 = T x_2 = x, ~ x \in X$.