Question

please proof and explain.

thank you

1. Let W be a finitely generated subspace of a vector space V . Prove that W has a basis.

2. Let W be a finitely generated subspace of a vector space V . Prove that all bases for W have the same cardinality.

Answer 1

1. Let W = span {w₁,w₂,…,w_n} be a finitely generated vector subspace of a vector space V.

If the set S = {w₁,w₂,…,w_n} is linearly dependent, then it is a basis for W. However, if the set S is linearly dependent, then there exist scalars a₁,a₂,…,a_n, not all zero , such that a₁w₁+a₂w₂+…+a_nw_n = 0.

Let a_i ≠ 0. Then a_iw_i = -( a₁w₁+a₂w₂+…+a_i-1w_i-1 + a_i+1w_i+1 +…+ a_nw_n ) so that w_i = -(a₁w₁/a_i +…+ a_i-1w_i-1 a_i + a_i+1w_i+1 / a_i +…+ a_nw_n /a_i). This means that w_i is a linear combination of w₁,w₂,…,w_i-1, w_i+1,…,w_n.

Now, if the set { w₁,w₂,…,w_i-1, w_i+1,…,w_n } is linearly dependent, then it is a basis for W. If not, then we keep repeating this process till we get a linearly dependent subset of S. Then this subset of S is a basis for W.

Please post the second question again, separately.