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Let $V$ be a vector space and $U,W$ and $Z$ subspaces of $V$.

Does $V = W \oplus Z$ and $V = U \oplus Z$ necessarily means that $W = U$?

Thanks!

gymbvghjkgkjkhgfkl
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user401575
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  • Why would you call such a property transitivity? It would be called something like a 'cancellation law for direct sums'. But as the answer points out, it is false anyway. – MooS Mar 09 '17 at 08:55

1 Answers1

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No, for example $\mathbb{R}^2 = \operatorname{span}((1,0)) \oplus \operatorname{span}((0,1))$ and $\mathbb{R}^2 = \operatorname{span}((1,0))\oplus \operatorname{span}((1,1))$.

Jonathan Grant
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