Abstract Linear Algebra Question See attached image Let V Eq

Abstract Linear Algebra Question. See attached image.

Let V Equivalence R^3 and let U be the subspace spanned by A Equivalence {(-3, -2, 0), (4, -1, 2)}. Is there a subspace W of V such that all of the following hold: W Not subset U dim (U) + dim(W) > dim(V) dim(U + W)

Solution

If V = R³ , them dim(V) = dim (R³) = 3 . Also dim U = 2 . We are given that W is not a subspace of U . Let W be a subspace of V with dimension w.

Then since dim (U) + dim(W) > dim (V) , we have 2 + w > 3 or, w > 1... (1)

Also, since dim(U+W) < dimV , we have dim(U+W) < 3 ...(2)

If equation 2 is to be satisfied, dim(U+W) has to be 2. However since dim (U) = 2, W can at best be an empty set unless  W is a subspace of U. But it is given that W is not a subspace of U ( then dim (U + W) = DIM ( U ) = 2 as U +W = W). Therefore, W is an empty set. This contradicts the 1st equation ( w > 1.) Thus, is W is not a subspace of U, then the given two conditions cannot be satisfied together. Thus no such W exists.

Now if W is an epty set, then dim (W) = 0. i.e w = 0.