Hi Im sort of new to vector subspaces could anybody help me with this question.
Let $U$, $V$, $W$ be subspaces of $\mathbb{R}^2$ such that $U + V = U + W$. Does it follow that $V = W$?
Let $U$, $V$, $W$ be subspaces of $\mathbb{R}^2$
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0HINT: Let both $V$ and $W$ be subspaces of $U$. – 2017-02-14
2 Answers
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No, there are some counterexamples you should try out. Consider the case in which $U + V = \mathbb{R}^2$ and $U$ and $V$ are independent lines through the origin.
If $U + W = \mathbb{R}^2$, then $W$ could be a line through the origin which is different from both $U$ and $V$.
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Consider $U = V = \{(x,0) \vert x \in \mathbb{R} \}$ and $W = \{(0,0)\}$. Then $U + V = U + W$ but $V \neq W$.