A Linear Algerbra Question Regarding Span, Linear Independence and Base

[The Student]

My notes have, "If V is spanned by m linearly independent vectors, we would like to define the dimension of V to be equal to m. The problem is this: what if V has two different bases, one with m vectors and another with k vectors where k does not equal m?".

Then after another page of notes, they go to explain, "Theorem. Any n linearly independent vectors in R^n span R^n"

I don't understand what the issue is against equating V to have m dimensions. The first sentence says that V is spanned by m linearly independent vectors. How could k vectors possibly change V from having m dimensions? And how could there be a V with multiple bases?

Does the theorem nullify the problem in the first part of the notes?

 

[Bob Brown MSEE]

Yes m.
Impossible if k<m
so k>m and not linearly independent if basis of k vectors are contained in V

 

[The Student]

Yes m.
Impossible if k<m
so k>m and not linearly independent if basis of k vectors are contained in V

Thank-you