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If I have a vector space $U$ of finite dimension and two subspaces $V$ and $W$ such that $\dim(U)=\dim(V)+\dim(W)$, then it is not necessarily true that $U=V\oplus W$, right?

For example, if in $\mathbb R^3$ I have a plane $\pi$ that passes through the origin and I have a line $r$ contained in $\pi$ that also passes through the origin, then $r$ and $\pi$ are subspaces of $\mathbb R^3$ such that $\dim(r)+\dim(\pi)=\dim(\mathbb R^3)$ but $\mathbb R^3\neq r\oplus \pi$.

Are there special cases in which the fact that the vector subspaces have complementary dimensions implies that they are also complementary subspaces?

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1 Answer 1

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Let $V,W$ be subspaces with $\dim V + \dim W = \dim U < \infty$.

Then $V \oplus W = U$ iff $V \cap W = 0$.

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