Cancellation law for direct sums

linear-algebra direct-sum

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!


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

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