Proof that W is uniquely determined by $M = \{v = x + w : w \in W\}$ and $x_{1} - x_{2} \in W_{1} \cap W_{2}$

Question

Let $M = \{v = x + w : w \in W\}$ linear manifold, where $x$ is a vector from a linear space $V$, and $W$ is a subspace of $V$. In one textbook, there is a proof that W is uniquely determined by M and it looks as follows.

Let $M = x_{1} + W_{1} = x_{2} + W_{2}$ then $x_{1} - x_{2} \in W_{1} \cap W_{2}$. Let $y \in W_{1}$ then $y + (x_{1} - x_{2}) \in W_{2} \Rightarrow y \in W_{2}$ then $y + (x_{2} - x_{1}) \in W_{1} \Rightarrow y \in W_{1}$.

I can't understand why $x_{1} - x_{2} \in W_{1} \cap W_{2}$ and $y + (x_{1} - x_{2}) \in W_{2}$.

Could you please give me a little hint about what's going on here?

Answer 1

• Since $x_1=x_1+0\in x_1+W_1=x_2+W_2,$ we have $x_1-x_2\in W_2.$ Similarly, $x_1-x_2\in W_1.$ But this first point is actually useless.
• From $W_1+x_1=W_2+x_2,$ we derive directly $W_1+x_1-x_2=W_2$ hence if $y\in W_1$ then $y+x_1-x_2\in W_2.$

A more informative proof of this uniqueness is: if $M=x+W$ where $W$ is a linear subspace, then $W=M-M,$ i.e. $$W=\{u-v\mid u,v\in M\}.$$