5. Prove the Following:

a. Let {v1, . . . , vn} be a finite collection of vectors in avector space V and suppose that it is not a linearly independentset.

i. Show that one can find a vector w ? {v1, . . . , vn} suchthat w ? Span(S) for S := {v1, . . . , vn} {w}. Conclude thatSpan(S) = Span(v1, . . . , vn).

ii. Suppose T ? {v1, . . . , vn} is known to be a linearlyindependent subset. Argue that the vector w from the previous partcan be chosen from the set {v1, . . . , vn} T.

b. Let V be a vector space and v ? V a vector in it. Argue thatthe set {v} is a linearly independent set if and only if v 6= ~0.Then use this fact together with part i of part a to prove that if{v1, . . . , vn} is any finite subset of V containing at least onenon-zero vector, you can obtain a basis of Span(v1, . . . , vn) bysimply discarding some of the vectors vi from the set {v1, . . . ,vn}.

c. Suppose {v1, . . . , vn} is a linearly independent set in Vand that {w1, . . . , wm} is a spanning set in V.

i. Prove that n ? m. Hint: use part ii of part a to argue that,for any r ? min(m, n), there is a subset T ? {w1, . . . , wm} ofsize r such that {v1, . . . , vr , w1, . . . , wm} T is aspanning set. Then consider the two possibilities when r = min(m,n).

ii. Conclude that if a vector space has a finite spanning set,then any two bases are finite of equal length. (Necessarily, thismeans that our notion of dimension from class is well-defined andany vector space with a finite spanning set hence has finitedimension).