Theorem:
The
and have the same dimension.(In spite of the size of )
Proof:
Let
be the RREF of
Number of leading1's variables
Def of rank():
and nullity(零化度) of , is the solution set of
EX: 
Theorem(矩陣rank等於與leading1數):
If
is a m-by-n matrix,then <a>
number of leading1's variables in the general solution of <b>
number of free variables in the general solution of
Theorem(
If
is a matrix with columns, then
Proof:
Since
has n columns, it implies that the homogeneous linear system has n unknowns. These fall into two distinct categories: the leading1's variables and free variables. Thus:
number of leading1's variables + number of free variables = n
number of leading1's variables = number of linearly independent column vectors of and number of free variables of
the number of free parameters in the general solution of Substitute
into we have:
Theorem(Equivalent statement):
If
is a n-by-n matrix, then the following statements are equivalent <a>
is invertible <b>
has only the trivial solution <c> The RREF of
is <d>
is expressible as a product of elementary matrices <e>
is consistent for every n-by-1 matrix <f>
has exactly one solution for every n-by-1 matrix <g> det(A) = 0
<h> The column vectors of A are linearly independent
<i> The row vectors of A are linearly independent
<j> The column vectors of
span <k> The row vectors of
span <l> The column vectors of
form a basis for <m> The row vectors of
form a basis for <n>
<o>