1. Use the theorem which says that

. Notice that if U and W are different then U+W is bigger than either U or W. Since U and W have dimension n–1 it follows that U+W must be the whole of V.

2. This result is false. For example, take V to be 2-dimensional space

with the standard basis consisting of

and

, and let U be the 1-dimensional subspace spanned by (1,1).

3. This result is true. If rank(B) = 2 then the subspace

is 2-dimensional. But the null space of A is only 1-dimensional. So the space

cannot just consist of the zero vector.