Let

be a spanning set of V.

Prove that for every

, the set

is a lin. dep spanning set of V.

Seems trivially but not sure how to say it correctly.

If S isn't a lin ind set of vectors, some vectors in S can be removed to form a minimal spanning set say S' of basis vectors of v. If S is lin. ind., then S is a set of basis vectors for V.

By definition of span, any vector in V can be written as a linear combination of vectors in S or S' depending if S is lin ind. It seems to follow trivially. Is there a better way to say this?