Effectively I'm trying to show that: $Q \subset R$ and $R \subset Q$. We know that $P+Q \subset P+R$ and that $P+R \subset P+Q$ and I'm really struggling to get from this to a solution. I have been staring at this problem for hours now, any hints would be greatly appreciated.
Let $P,Q,R$ be subspaces of $\mathbb{R}^2$ such that $P+Q=P+R$, is it the case that $Q=R$?
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linear-algebra
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2You need some assumptions. If, say, $P=\mathbb R^2$ then any $Q,R$ will work. Or, for general $P$, take $Q=P$ and $R=\{0\}$. – 2017-02-14
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2And other nontrivial cases are easy to obtain such as $P$ generated by $(1,0),$ $Q$ generated by $(0,1)$ and $R$ generated by $(1,1).$ – 2017-02-14
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It''s trivially false. Otherwise, it would mean that, say in $\mathbf R^2$, if two bases share a vector, the other vectors are collinear.