Dimension of the sum of three subspaces We know that $$\dim(U_1 + U_2) = \dim U_1 + \dim U_2 - \dim(U_1 \cap U_2)$$ if $U_1$ and $U_2$ are finite dimensional subspaces.
For three finite dimensional subspaces prove or give a counterexample for the following:
$$
\begin{align}
\dim(U_1 + U_2 + U_3) &= \dim U_1 + \dim U_2 + \dim U_3 \\
&- \dim(U_1 \cap U_2) - \dim(U_2 \cap U_3) - \dim(U_1 \cap U_2)\\
&+ \dim(U_1 \cap U_2 \cap U_3) 
\end{align}$$
Its basically the formula for the union of three sets. I think that this is false but the only reason I can think it would be is because the union of subspaces is rarely a subspace itself, even though it applies to sets, so it seems to me like this formula which works for sets shouldn't work for subsets because it would disrupt overlapping elements of $U_1 + U_2 + U_3$ and run into complications with the sum being closed under addition or something like that. Help much appreciated thank you!

That is not true. But, for 3 linearly independent spaces, it is true,and can be generalized by the inclusion–exclusion principle, to any finite number of spaces.
