If S spans a vector space V, but S is not linearly independent in V, then there exists a vector v in S such that the set difference also spans V.

Question

If S spans a vector space V, but S is not linearly independent in V, then there exists a vector v in S such that the set difference  also spans V.

Quizplus · Accepted Answer

Since S is not linearly independent, there exists a vector v in S that can be expressed as a linear combination of the other vectors in S. Let's call those other vectors S'. Since v can be expressed as a linear combination of S', we can remove it from S without changing the span of S, since any vector that v could have contributed to the span can still be expressed as a linear combination of S'. Thus, the set difference between S and {v} (which is just S') still spans V.

If S Spans a Vector Space V, but S Is