Suppose S_1={u_1,u_2} and S_2={v_1,v_2} are each independent sets of vectors in an n-dimensional vector space V.

Question

Let us assume that every vector in S_2 is a linear combination of vectors in S_1.

Question: Does that mean that S_1 and S_2 are bases for the same subspace of V?

I know that the answer to this question is yes, subspaces spanned by both S_1 and S_2 are the same. Let's call them W_1 and W_2 respectively. How do we prove W_1=W_2? Equal subspaces when regarded as sets, must have the same elemnts. How can we show that EVERY vector in one subspace is also in the other subspace?

Answer 1

You'll need to ask yourself what an arbitrary member of W₁ looks like and what an arbitrary member of W₂ looks like. Then, follow these steps:

 

1. Prove that the arbitrary member of W₁ is in W₂. This means that W₁ is contained in W₂.

 

2. Prove that the arbitrary member of W₂ is in W₁. This means that W₂ is contained in W₁.

 

3. Since each set is contained in the other, the sets are equal.

 

 

Hope this helps! It's been a while since I've taken Linear Algebra; I wish I could give you a more detailed answer.