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I'm having trouble understanding a proof of the following theorem which allows it to be shown that all bases for a vector space have the same number of vectors, and that number is the dimension of the vector space (as you probably already know):

**1. The problem statement, all variables and given/known data**

Theorem:

If S = {v1, v2, ... , vn} is a basis for a vector space V and T = {w1, w2, ... , wk} is a linearly independent set of vectors in V, then k < n.

**2. Relevant equations**

Here's the proof: (sorry about all the text)

If k > n, then we consider the set

R1 = {w1,v1, v2, ... , vn}

Since S spans V, w1 can be written as a linear combination of the vi's.

w1 = c1v1 + ... + cnvn

Since T is linearly independent, w1 is nonzero and at least one of the coefficients ci is nonzero. Without loss of generality assume it is c1. We can solve for v1 and write v1 as a linear combination of w1, v2, ... vn. Hence

T1 = {w1, v2, ... , vn}

is a basis for V. Now let

R2 = {w1, w2, v2, ... , vn}

Similarly, w2 can be written as a linear combination of the rest and one of the coefficients is non zero. Note that since w1 and w2 are linearly independent, at least one of the vi coefficients must be nonzero. We can assume that this nonzero coefficient is v2 and as before we see that

T2 = {w1, w2,v3, ... , vn}

is a basis for V. Continuing this process we see that

Tn = {w1, w2, ... , wn}

is a basis for V. But then Tn spans V and hence wn+1 is a linear combination of vectors in Tn. This is a contradiction since the w's are linearly independent. Hence n > k.

**3. The attempt at a solution**

Where I get lost is where it says that T1 is a basis for V. Just because one of the vectors from the basis can be written as a linear combination of T1, why does that mean that T1 is linearly independent and that it spans V? When the author writes that we can assume it's v1 without loss of generalization, I guess that he/she means that we could pick any other vector in S, but that still doesn't imply (as far as I can tell) that T1 spans V

Any help will be greatly appreciated