If S is a linearly independent subset of a vector space V, but S does not span V, then there exists a vector v in V such that the set is also linearly independent.

Question

If S is a linearly independent subset of a vector space V, but S does not span V, then there exists a vector v in V such that the set  is also linearly independent.

Quizplus · Accepted Answer

Since S does not span V, there exists a vector v in V that is not in the span of S. Thus, v is linearly independent from the set 11edaba5_20a4_adbf_a975_7581548987c6_TBMC1039_00, which consists of S plus an extra vector v. Therefore, the set 11edaba5_20a4_adbf_a975_7581548987c6_TBMC1039_00 is also linearly independent.

If S Is a Linearly Independent Subset of a Vector