Question

Suppose U and W are subspaces of a vector space V, where dimR(V)=8. Suppose dimR(U)= 4, and dimR(W)= 5. What are the possible values of dimR(U∩ W)?

Answer 1

$dimR(V)=8\\ dimR(U)= 4\\ dimR(W)= 5$

We know that

$max \{0, dimR(U) +dimR(W) -dimR(V)\}\leq \\ \leq dim \space R (UnW) ≤ min (dimR(U) ,dimR(W)),\\ max \{0, 4 +5 -8\}≤dimR (UnW) ≤min(4,5),\\ max \{0, 1\}≤dimR (UnW) ≤min(4,5),\\ 1≤ dimR (UnW) ≤4\\$

Hence , the possible values of dimR(UnW) is 1, 2, 3, 4.