Hello all I was given this question in linear algebra it is two parts and asks to prove or give a counterexample. We are given a vector space V and a subspace of it W and the quotient map $ \pi : V \to V/W $ are asked:
- if $ V = V_1 \oplus V_2 $ then $ V/W = \pi(V_1) \oplus \pi(V_2) $
- if $ V/W = V_1 \oplus V_2 $ then $ V = \pi^{-1}(V_1) \oplus \pi^{-1}V_2 $
I have tried but I can neither prove or come up with a counterexample for any of them. Can anyone please help me out with this? Thank you all
Counter example for the first:
In ${\mathbb{R}}^2$, let $V_i=$ span$(e_i)$, and $W= \{c(1,1)^t\}$ with $c \in \mathbb{R}$. Then $e_i + W \in \pi(V_i)$, and $(e_1+ W) = (-e_2 + W)$ (visually obvious), so the intersection is not trivial, and $\pi(V_1)$ and $\pi(V_2)$ are not independent.
For 2), let $v_i \in \pi^{-1}(\pi(V_i))$, and assume $v_1+v_2= 0$. Then also $$(v_1+W)+(v_2+W) = 0+ W$$ But as $\pi(V_1) \oplus \pi(V_2)$, the sum must be trivial, and so $v_1=v_2=0$. So we have independence. If $y\in V$, then also $y+W \in V/W$, and so $$y+W = (v_1+W) + (v_2 + W)$$ with $v_i +W \in \pi(V_i)$, and so $v_i \in \pi^{-1}(\pi(V_i))$. But then also $y= v_1 + v_2$ by the way cosets add in $V/W$. So we have $V$ is the sum of the $\pi^{-1}(\pi(V_1))$, and so also the direct sum.