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Is there a (valid) formula for $\dim(U + V + W)$? I know from MO that $\begin{align*} \dim(U + V + W) &= \dim(U) + \dim(V) + \dim(W)\\ &\qquad\mathop{-} \dim(U \cap V) - \dim(U \cap W) - \dim(V \cap W)\\&\qquad \mathop{+} \dim(U \cap V \cap W) \end{align*}$ is wrong.

Can we relate $\dim(U + V + W)$ with the cardinality of some of their quotient spaces? (sorry if this is a dummy question but I'm not any familiar with quotient spaces).

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    Presumably, Vicfred wants to write $\dim(U+V+W)$ as a linear combination of $\dim(U)$, $\dim(V)$, $\dim(W)$, $\dim(U\cap W)$, $\dim(U\cap V)$, $\dim(V \cap W)$, and $\dim(U \cap V \cap W)$.2011-04-25

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I don't think you can do better than this: $\begin{align*} \dim(U +V + W) &= \dim((U +V) + W) \\ &= \dim(U +V) + \dim W - \dim((U+V)\cap W) \\ &= \dim U + \dim V - \dim (U \cap V) + \dim W - \dim((U+V)\cap W) \end{align*}$ Now you're stuck with $\dim((U+V)\cap W) $, for which there does not seem to be a simple formula.

(BTW, see also https://mathoverflow.net/questions/17740/is-there-a-version-of-inclusion-exclusion-for-vector-spaces but you probably know about this already.)