Re: Linear Algebra Question (another one)

In article
<90da6086-9c08-41c3-adee-2374818edebd@[EMAIL PROTECTED]
>,
Citizen <FlammesSombres@[EMAIL PROTECTED]
> wrote:

> Hi, can anyone please help with this question?
> 
> If a matrix has r pivot columns, how do you know that its trans****t
> also has r pivot columns?
> 
> Thanks for your help

How you do this depends on what you know. Here is one way:

Let A be m x n. N( ) = null space; C( ) = column space; 
R( ) = row space; A' means the transpose of A.

n = dim(N(A)) + dim(C(A))
  = (n - r) + r
  = (n - r) + dim(R(A)).
  
On the other hand, let s be the rank of A'.

m = dim(N(A')) + dim(C(A))
  = (m - s) + s
  = (m - s) + dim(R(A'))
  = (m - s) + dim(C(A))
  = (m - s) + r
  
  So, s = r.

-- 
Paul Sperry
Columbia, SC (USA)