We know that if $U$ is a subspace of $V$ (finite-dimensional), then $V = U \oplus U^\perp$. Given this theorem, how does this lead to the conclusion that $\dim U^\perp = \dim V - \dim U$? This seems like a stupid question because it seems obvious, but I'm still unclear with this notion of a direct sum and the dimensions of subspaces adding up to the vector space.
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1$\begingroup$ $\dim V=\dim U + \dim U^\perp$ $\endgroup$– J. W. TannerNov 16, 2020 at 4:27
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1$\begingroup$ Cf. this question $\endgroup$– J. W. TannerNov 16, 2020 at 4:28
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