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?
to amplify: we can write (because {v1,...,vn} spans V) w = a1v1 + a2v2 +...+ anvn.
therefore w - a1v1 - a2v2 - ... - anvn = 0. note that the coefficient of w is 1 ≠ 0, so we have found a linear combination
c1w + c2v1 + ...+ c(n+1)vn = 0 with not all the cj = 0. so {w,v1,...,vn} is linearly dependent.
you don't need to find the subset S'. all you need is the fact that S spans V, so you can express w as a linear combination of the vj.
what you did is take S, create a basis S' and then show {w}U S' was linearly dependent. you don't need the middle part, adding anything to a spanning set S
creates a linearly dependent set, whether or not S was linearly independent.