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
(1)
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
(2)
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
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