I am given a vector space $V$ of dimension $n$ and a subspace $S$ of dimension $w$. Is it possible to find the dimension of $S^{\perp}$ without actually finding $S^{\perp}$? I mean is there any formula ?
A simple dimension of a subspace problem
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linear-algebra
vector-spaces
orthogonality
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0In the finite-dimensional setting certainly yes. – 2017-02-08
1 Answers
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Yes, if $V$ is a finite-dimensional vector space and $S$ is a subspace of $V$ then $V=S\oplus S^{\perp}$, hence $\dim S^{\perp}=\dim V-\dim S$.