I am trying to find the smallest possible dimension of a vector space V, which has 3 subspaces $S_1, S_2, S_3$. These subspaces have the dimensions of: $dim(S_1) = 4dim(S_2) = 6dim(S_3)$. It is also known that $S_1 \cap S_3 = \{0\}$ and that $S_2\neq\{0\}$. I am not really sure how to approach this question. But I thought that maybe using this might be a way to start:
$dim(U) + dim(V ) − dim(U ∩ V ) = dim(U + V )$. For the relationship between $S_1, S_3$
To get: \begin{eqnarray} dim(S_1) + dim(S_3) - {0} = dim(S_1 + S_3)\\ dim(S_1) = 6dim(S_3)\\ 7dim(S_3) = dim(S_1 + S_3)\\ \end{eqnarray} But I am unsure if this is the right track of working or even how to bring in $S_2$ into this.