Proposition. In any finite-dimensional vector
space the number of elements in any linearly independent
sequence is at most equal to the number of elements in a
given spanning set.
Proof. Let the vector space be spanned by
, and let be a linearly
independent sequence. The task is to show .
Since is a spanning set, one has
for each , . One may express this very
concisely by writing
where is the row
of elements of ,
is the row
of elements of ,
and is the matrix
.
If , then the reduced row echelon form of can have
at most non-zero rows and, therefore, at most pivot
columns. So at least one column of is not a pivot column.
This means that column is a linear combination of the pivot
columns to its left. Hence, if is the column of coefficients
of the ensuing linear relation with ,
then one has
which means that cannot be linearly independent,
a contradiction made possible by assuming . Hence
.