Related to basis Prove or give a counterexample If v1 v2 v3

Related to basis.

Prove or give a counterexample: If v_1, v_2, v_3, v_4 is a basis of V and U is a subspace of V such that v_1, v_2 U and v_3 U and v_4 U, then v_1, v_2 is a basis of U.

Solution

Proof.

Let v1,v2 not be a basis of U

But since v1,v2 is part of basis of V so v1,v2 are linearly independent and hence it means we need to extend the set v1,v2 to form basis for U is add more vectors to this set so that the set spans U and is linearly independent.

So the set will ahve a minimum of 3 and maximum of 4 vectors

Case 1. Basis for U has 4 vectors

But V itself has 4 vectors in basis and a proper subspace of V cannot have same number of vectors in basis. ANd U is a proper subspace because the vectors v3,v4 are not in U. So basis for U cannot have 4 vectors

Case 2: Basis for U has 3 vectors

v3,v4 are not in U so they are not in span of these three vectors. Hence we can extend this set by adding v3,v4 and get a set of linearly independent 5 vectors in V which is not possible as V can have atmost 4 linearly independent vectors

Hence, basis of U has 2 linearly independent vectors

And v1,v2 are linearly independent and hence form basis for U