Write two non trivial subspaces V and W of R5 of dimension 3

Write two non trivial subspaces V and W of R5 of dimension 3. What is V W, the intersection of V and W. ( Recall that V W is the set of all elements common to both V and W.) ( Your answer will depend on your examples for S and T).

Solution

The standard basis for R⁵ is { (1,0,0,0,0)^T, (0,1,0,0,0)^T, (0,0,1,0,0)^T, (0,0,0,1,0)^T, (0,0,0,0,1)^T}. Let V = span { (1,0,0,0,0)^T, (0,1,0,0,0)^T, (0,0,1,0,0)^T} and W = span{(0,0,1,0,0)^T, (0,0,0,1,0)^T, (0,0,0,0,1)^T} be two subspaces of R⁵ . Then V W = span{(0,0,1,0,0)^T}.