Set of subspaces of vector space

Let $V$ be a (real) vector space. Consider the set $mathscr{V}$ consisting of all subspaces of $V .$ Equipping $mathscr{V}$ with the addition of subspaces, is it possible to define a notion of scalar multiplication on $mathscr{V}$ so that it becomes a vector space?

This seems very interesting question
Set of All subspaces of vector space V form vector space or not ?
As we know sum will be easy to check since if W and Z are subspaces of V then W + Z is also subspace. But what about scalar multiplication.