Is it true? $\dim(U+W)=\dim(U\cap W)+1$ implies $U\subseteq W$

Question

I'm curious whether or not this statement is true.

If $U,W$ are finite-dimensional space satisfying $\dim(U+W)=\dim(U\cap W)+1,$ then $U\subseteq W.$

Any help would be appreciated!

Thank you

Answer 1

Suppose not. Then $$\dim(U+V)\geq \max(\dim(U),\dim(V))+1>\dim(U\cap V)+1$$ the first inequality follows from the fact that there is one vector in each of the two spaces which does not belong to the span of the other, so the dimension of the cumulative span becomes larger.

The second inequality is strict because of the same reason: the intersection must be of dimension strictly less than the maximal dimension of the two spaces.

---

EDIT: now that we know that either $U\subseteq V$ (or conversely), we note that $$U+V=V\quad \text{and}\quad U\cap V=U.$$ thus the equality becomes $$\dim(V)=\dim(U)+1$$