Hi I need some help understanding this theorem,thanks

I dont understand this fully ..especiallly the bit about a contradiction ..what is the implication of this theorem??

Let w be a subspace of a vector space V and dim(V) =n

then i)dim(W)≤n;

and

ii)dim(W)=n iff W=V

Proof) i) any basis B of W must contain at most n vectors ,since otherwise B is Linearly dependendent by theorem(*). Hence dim(W)≤n.

ii)If W=V then dim(W)=dim(V)=n

conversely suppose that dim(W)=n and let B={w,,w2,...wn} be a basis of W.

Suppose for a contradiction that W≠V.Then there is v∈V such that v does not exist in W ..

a0v+ a1w1+a2w2+...anWn=0

for some scalars a0,a1,....an

then a0=0 since otherwise..

v=[-1/(a0)][ ]∈Span(B)=W

so a1w1+a2w2+...anWn=0

a1w1+a2w2+...anwn=0

a1=a2-...=an=0

since B is linear independent

this shows X={v1,w1,....,wn} is linear independent,which is impossible since X contains n vectors ,so X is Linearly depenedent by theorem(*)

ehence W=V.

(*)theorem

Let X={v1,v2,....Vn} be a basis of a vector space V and Y={w1,w2,...Wm} a subset of V.

i)If m>n then Y is linear dependent..

ii)if m<n then Y does not span V.