Correctness of the space dimension definition
This proof has been taken from I. M. Gelfand, Lectures in Linear Algebra, 4th edition, 1970 (in Russian).
Lemma. Let be some vectors. Suppose that vectors are linearly independent and belong to the span Then
Proof. We prove the lemma by induction. Let Then (linear dependence for an empty set of vectors is not defined) and the inequality is trivial.
Suppose the statement holds for and let us prove it for Since we have
If all of are zero, we have and then by the induction assumption and, trivially,
Thus, we can assume that not all of are zero. Suppose Then we can solve the last equation in (1) for
Plugging this equation in the first equation in (1) we get
Send to the left side; the remaining expression on the right side is a linear combination of ; exact expressions of the coefficients of this linear combination don't matter. The result will be After doing the same with the first equations of (1) we get the system
This shows that the vectors
belong to If they are linearly independent, we can use the induction assumption to conclude that which will prove
Suppose with some
By the assumed linear independence of this implies so the system is linearly independent.
Theorem. The definition of the space dimension is correct.
Proof. We write if a) contains linearly independent vectors and b) We need to prove that any system with properties a) and b) has the same number of vectors. Suppose is another such system. Since belong to by the lemma Similarly, So