Echelon Test for Independent Sets
Theorem:
Assume
!
!
,
...,
!
"
are vectors in
ℝ
#
.
Let A be the matrix with columns
!
!
,
...,
!
"
.
Let B be a matrix equivalent to A that is in echelon form.
Then
!
!
,
...,
!
"
are
independent
if and only if B has a pivot in every
column
.
Why?
If B has a pivot in every column, then A
x
=0 has a unique solution.
So, by definition of independent sets
!
!
,
...,
!
"
are independent.
On the other hand, ...If B has a column with no pivot, then A
x
=0 has a
free variable.
Therefore, it will have an infinite number of solutions so
!
!
,
...,
!
"
are dependent.
