b1 is a basis for W1 and b2 is a basis for W2 Please prove t

b1 is a basis for W1 and b2 is a basis for W2.

Please, prove that if intersection of b1,b2 is empty set, intersection of W1,W2 is zero vector.

Thank you.

Solution

for if dim(V) 2, any basis bi for W has at least two elements, say b1 and b2.

so span(W1 - {b1}) is a subspace of V that does not include b1 (since b1 is LI from the other basis vectors), and since W1 - {b1} is not empty (it contains at least b2), dim(span(W1 - {b1})) 1, so this is not the 0-subspace (which has dimension 0).

so intersection of W1,W2 is zero vector.