If $U_1$ and $U_2$ are subspaces of a finite dimensional vector space, then $$\dim(U_1+U_2) = \dim U_1+\dim U_2-\dim(U_1 \cap U_2).$$
How can one generalize this notion to a collection of $n$ subspaces $U_1,\ldots,U_n$?
Or what does $\dim(U_1+\cdots+U_n)$ equal?
Best Answer
We can write $U_1+U_2+\cdots$ as the sum of two subspaces, for example $U_1+(U_2+U_3+\cdots)$, and then: \begin{equation}\dim(U_1+U_2+\cdots)=\dim(U_1)+\dim(U_2+U_3+\cdots)-\dim(U_1\cap(U_2+U_3+\cdots)).\end{equation} So then applying the same argument to the term $\dim(U_2+U_3+\cdots)$ recursively you can then eventually arrive at an equation consisting of the sum of the dimensions of each subspace minus a bunch of nasty terms involving intersections similar to $\dim(U_1\cap(U_2+U_3+\cdots))$.